teaching calculus today in college

[mathwonk]

The biggest task I have seems to be helping students learn how to learn. Some fail to come to class, others never look at the notes they take, and many seem not to even open the book.

Many never ask questions, and those who do, often ask things that could be found immediately by looking them up in the index of the book. People who ignore office hours for weeks expect me to schedule extra help sessions the day before the test. Questions more often focus on "what will be tested?" instead of how to understand what has been taught.

Everyone seems to have taken calculus in high school, but most also seem not to know anything about algebra or geometry or trigonometry. With the advent of calculators some also do not know simple arithmetic, like how to multiply two digit numerals. (I have had students who had to add up a column of thirteen 65's on a test, apparently not knowing how to multiply 13 by 65.)

Many think that having taken a subject "2 years ago" is a valid excuse to have forgotten the material, and to expect the teacher to reteach the prerequisites. Appparently no one ever dreams of reviewing the prerequisites before the course starts. Books like "Calculus for cretins" are apparently more popular than books like "Calculus for science majors".

When I was in college students like this were just ignored, or expected to flunk out, but in today's setting there are so many like this that they form the primary market. With all good faith to teach these stduents, the failure rate is still about 50% in college calculus across the nation, in my opinion. What are some ideas on how to improve this?

[Muzza]

When I was in college students like this were just ignored, or expected to flunk out


How many people took calculus way back when you were in college (compared to today)?

[shmoe]

I hear your pain. My favorite are students who fail the course, then beg to be passed when they've managed to miss half to term work and I have absolutely no idea who they are apart from a name and number on the class list. Where were they all year? Somewhere along the line students developed a sense of entitlement, they (or mommy and daddy) are paying big bucks for tuition so they somehow deserve good grades no matter what they do.

The most important thing I try to drill into students heads is the only way to learn mathematics is to do mathematics. The first step to this is giving problems an honest attempt before giving up. This means trees will die. I'd rather see a student come in with a page of nonsense that failed to work than a blank page and expression. They're often afraid to make mistakes so they don't even try. This is rubbish, the number of mathematics mistakes I can make in a day is limited only by the number of hours I'm awake. I try to lead by example here and show them two of my favorite textbooks, which are just problem books (one algebraic number theory, the other analytic) and explain the mounds of paper I've burned through over the years.

It takes work, but if they are willing to put in the time to understand the course, I'm willing to put in the time to assist. I have nearly infinite patience for students who are obviously doing the work. Others, not so much

[mathwonk]

Muzza, I do not have figures, but it seems many fewer took calculus then, and many more took algebra, etc, in high school. When I started college I only knew algebra and geometry, not even trig. I believe the change in high schools from teaching precalculus subjects thoroughly, to offering too many people watered down calculus today without adequate background, is a big part of the problem, but I am not trying to place blame, just think of solutions.

In fact I was one of the students in college who flunked out from poor study habits myself. (I was in a lecture class of 135 students that met Tues, Thur, and Sat, at 9am, not always including me.) In my case working in a factory helped give me an attitude adjustment. When I got back in college, I was not allowed to repeat anything, but had to pick up where I had left off. (We were admitted for 8 semesters, no more. The philosophy was: either graduate, or leave without a diploma so someone else could have the spot you are wasting.)

I got a D on my first test back, in diff eq, after not learning calculus. When I complained to the teacher I was being penalized for stuff from the previous course he just said "well, mathematics is cumulative". So I got a Schaum's Outline and began burning up the scratch paper as recommended by Shmoe. I ended with an A+.

The next year I asked the professor teaching honors advanced calc what I needed to get into his course. I listened, got a copy of Widders Advanced calc and read what he recommended over the summer. I managed a B+ and an A- in a course that covered Banach space calculus, infinite dimensional spectral theory of compact self adjoint operators, and differentiable manifolds.

By senior year I was in graduate real analysis and holding my own.

[mathwonk]

One thing I think does work, is the patience shown by people on this forum, at helping people without doing their work for them.

[Tide]

Mathwonk,

The part that gets me is how many students will flat out refuse to read the textbook. Some who claim to have read it will simply declare it makes no sense as if it's YOUR fault and then expect you to magically impart the knowledge in to their brains.

Along the same lines I can't count the number of times when posing a problem some student will simply demand that you tell them the answer or show them how to do it without even the slightest effort on their part. It's as if when you finally get around to testing them they expect the same question to be asked and all they need to do is provide the answer they've already seen. It's a shock to them when they encounter new problems on a test and then they complain bitterly that you never showed them how to do that!

Of course, none of that changes their mindset as the course progresses.

[mathwonk]

Tide, you remind me of the student who complained that I asked him to maximize the volume of a closed top box, when in class I had only shown how to do it for an open topped box.

This may suggest again that I need to be understanding how to help my students broaden the scope of what they are "learning". Simply handing out a syllabus that says "you will be tested on your ability to use what you have learned in new situations" is not sufficient.

Perhaps we should accept that the frustrating experience of hearing the complaints about our tests is actually a learning experience for the student, as painful as it is for both of us, and just stick to our guns.

[modmans2ndcoming]

you know why no one knows their algebra? because Math is not taken seriously enough and the methods used do not teach vocabulary so it is like trying to use a hammer and nail to put two boards together but no knowing what the name of either are. heck, even if you said a name of an algebraic tool to me today, there is a good chance I will not know it from that descriptor, but I do algebra like it was second nature.

usually after calc 1 and 2 the students who are asking a lot of algebra questions have dropped and the ones that are left either know what they are doing or have low confidence so they ask. I found it helpful when my prof said "it is just paper, if it does not work out...ERASE IT!"

[modmans2ndcoming]

if you ask me, all students entering college need to be required to take Trig and pre calc there even if they test into calc. then the math department can know what to expect from the students in calc.

[modmans2ndcoming]

umm, isn't the volume of an open top box going to be the same as a closed top box if all parameters for the rest of the box the same?

seems high school needs to teach common sense as well.

[modmans2ndcoming]

Tide,

in the defense of many students, some of the calc books are just plain badly written. my Real Number Analysis book was more interesting than my calc book.

I think that calc book writers need to use less brevity in examples because I know that a lot of students tend to get lost in the details because they cannot figure out how the writer went from step one to step two. they could at least have an appendix with a full description of the example, step by step. that way the bigger picture will not be lost.

[HallsofIvy]

Tide, you remind me of the student who complained that I asked him to maximize the volume of a closed top box, when in class I had only shown how to do it for an open topped box.


I can top that! On a Calculus III test, I gave a question right out of the book (but not one that had been assigned) on finding the maximum temperature on a plate given the temperature as a function of x, y. When a student complained that we hadn't covered "that kind of problem", I pointed out that we had done a number of problems on finding the maximum of a function of two variables. He protested "they didn't have anything to do with temperature!"

On the other hand, once, when a student protested after a test that I hadn't taught them how to do "that kind of problem", I was able to point out that, not only was that specific problem one of the assigned homework problems (I do that with 1 or 2 problems on each test), but that a studeng had asked about it in class, we had gone over it in class, and I showed him where he had the complete solution in his notes!

Give us strength!

[Dburghoff]

As a freshman college student (in Calculus III), I'm kind of offended by the blanket statements flying around here. I spend at least a half hour every day studying from my notes and reading the next section (so that I may be able to participate intelligently in the next lecture), and that's in addition to my homework. I've never missed a single class, and I'm always prepared, as are most of the people in my class (as far as I can tell). I just think that many of you are displaying classic "kids these days..." syndrome. It's always easier to judge your juniors more harshly than you judge your peers.

On an unrelated note, the fact that many of you are teachers comforts me greatly. I often just read topics in various forums, and I'm always disturbed when you talk about things which I haven't yet covered, whether it be in physics or math. It makes me feel like I am behind, even though I know that I'm not.

[mathwonk]

The quality of calc books is another important point raised by modmans. Actually many calc books are excellently written when they first come out, but publishers push for dumbing them down, to raise sales, and they tend to decrease in quality as the later editions come out.

Everyone knows what the good calc books are; well written, and authoritative: Courant, Courant and John, Apostol, Spivak. These are the time - tested, great contemporary calc books, and they have held this position for many years.

Engineering problem solving is probably still best learnt from the original book of George B. Thomas, now sold as the "aternate edition".

By the way Modmans, here is the closed / open top box problem: given say 6 square feet of sheet metal with which to build a rectangular box with a square bottom, find the dimensions which maximize the volume if the box is to have a top, and also with no top.

Thus clearly you should be able to make a larger box with no top, than with a top, but also the dimensions are different, interestingly. See if you can imagine why. If you think about it and have some intuition, this does not even require calculus, but calculus does work on it.

Here is a recommendation of a good cheap, short, paperback calc book, the one by Elliot Gootman, selling new for about $15. Of course it does not contain all important topics, but it is well written with excellent clear explanations froma master teacher. And it is better to actually learn a few key topics, than walk around with a thick book one does not or cannot read.

If you really want to learn the subject thoroughly, then get one of the classics recommended above, and spend time with it.

[mathwonk]

I was afraid of what DBurghoof has said. Even though the statements here have clearly referred to "many" or "most" or even only one example, he takes them as directed at him. This is of course not the case, but it is unavoidable. I think if you will reflect on it Mr Burghoff, you will find that either you are at a very elite school, or you are a very unusual person at your school, and that indeed many students are not doing what you are doing. But nonetheless I apologize if you are offended. We are not worried about the future of students such as yourself. Those of you who actually go to class and prepare the lesson are the ones that make our job worthwhile.

Notice for example that all the professors on this site are donating their time, with nothing whatsoever to gain, largely because it is so rewarding to tutor interested students like yourself.

I might add however that 1/2 hour a day is not much study time for a genuinely challenging course. Most people agree that 2 hours per class hour is minimal. Perhaps you are one of the fortunate few who learn quickly and easily. It is also possible your class is too easy for your abilities. In the example I gave above of a class in which I went from a D to an A+ in one semester, notice I left that program afterwards and entered one in which I could not so easily earn such a high grade. I felt that those courses in which one earns an A+ are not sufficiently challenging.

[Hurkyl]

On an unrelated note, the fact that many of you are teachers comforts me greatly. I often just read topics in various forums, and I'm always disturbed when you talk about things which I haven't yet covered, whether it be in physics or math. It makes me feel like I am behind, even though I know that I'm not.

That feeling never goes away. :biggrin: No matter how much math I learn, I always encounter some new topic about which I feel my education cannot be complete without learning it!

[vladimir69]

One thing I think does work, is the patience shown by people on this forum, at helping people without doing their work for them.
i second that. the people here have always kindly replied to my questions in a non smart a$$ fashion and i greatly appreciate that - it gives me a warm fuzzy feeling all over

[mathwonk]

Well, perhaps this shuold have been obvious, but the key suggestions emerging seem to be:

1. Be patient.

2. Be clear.

3. minimize criticism.

[Tide]

As a freshman college student (in Calculus III), I'm kind of offended by the blanket statements flying around here. I spend at least a half hour every day studying from my notes and reading the next section (so that I may be able to participate intelligently in the next lecture), and that's in addition to my homework.

I'm certain no offense was intended by anyone here. Classes and students vary. I'm sure you've seen enough postings here to recognize that student capabilities and commitment vary immensely. You must recognize from what is written here that many participating students demonstrate extraordinary abilities but just as many are clearly over their heads in the courses they take. Both categories comprise your typical classmates but it's the latter group that has caught the attention of this thread.

[mathwonk]

Here is a very positive experience that happened recently. I gave a test and as a extra question asked students to pose and answer the most interesting question about the course material they could. This was a beginnning calculus course. Normally students either omit the question entirely or ask something trivial.

This time one student tried to solve the "ham sandwich" problem, that given any triangle and any line in the same plane, some translate of the line bisects the triangle by area! It blew me away. The student's understanding was very partial but did contain the essential idea of using the intermediate value theorem applied to the comtinuity of the areas with respect to the position of the line.

Afterward we chatted enjoyably about it and created another more elementary solution allowing one to actually solve for the position of the bisecting line, in principle.

This is a delightful experience which has happened only a few times in several decades, but is still wonderful. The moral dilemma now is whether to kick such a student out into a more advanced class or simply continue to enjoy their presence.

Of course sometimes students appreciate our dismissing them. I remember one of my students writing back after a several decades and thanking me for my classroom guidance, as apparently I inspired him to drop out of school and become a cartoonist!

[modmans2ndcoming]

With a top, the dimensions will be a cube 1ft x 1ft x 1ft

with out a top, each of the 5 sides should be 6/5ft x 6/5ft x 6/5ft.

this is just of the top of my head, no calc, just quick thinking. so if it is wrong, it is because I did not put to much thought into it.

[HallsofIvy]

As a freshman college student (in Calculus III), I'm kind of offended by the blanket statements flying around here. I spend at least a half hour every day studying from my notes and reading the next section (so that I may be able to participate intelligently in the next lecture), and that's in addition to my homework. I've never missed a single class, and I'm always prepared, as are most of the people in my class (as far as I can tell). I just think that many of you are displaying classic "kids these days..." syndrome. It's always easier to judge your juniors more harshly than you judge your peers.

Yes, I can see how you would be offended. We have many good student, even more mediocre students and a few that just drive us crazy! Sometimes you just have to laugh about them to blow off a little steam!

On an unrelated note, the fact that many of you are teachers comforts me greatly. I often just read topics in various forums, and I'm always disturbed when you talk about things which I haven't yet covered, whether it be in physics or math. It makes me feel like I am behind, even though I know that I'm not.

I've been teaching math for more years than I care to count but I see topics here that I am not at all sure about! Mathematics (not to mention physics) is a broad area and there is always more to learn.

[matt grime]

Such is the nature of teaching that you often tend not notice the good students until too late, so for the freshman calc major, sorry, but those who shout the loudest get the most attention. And they usually piss off the lecturer too.

Seeing as we're into our anecdotage:

A colleague from a previous workplace of a colleague at a place I no longer teach at decided to experiment with his classes. In one he taught as normal, in the other he assigned all the questions as homework that would eventually appear on the final, in the third he gave out the answers to the final before hand. The marks in each class were almost identical.

I had one student threaten to sue me because when I said:
"Yes, you failed the midterm by 2%, I don't see why you can't pull that round and pass the course though" took that to mean he could fail the final by 2% and still get a C. Apparently I was jeapordizing his future earnings.

Anyone see the old Onion article about 'new, principled teacher offended by older teachers ridiculing <some student's> Hamlet essay in staff room'?

Some times we do need to blow off steam. My bete noir is when students make the exact mistake on homework that I told them was the common error and if it seems strange that's where you've gone wrong. (Commonest one, working out the length of i-j, say, to be zero).

[mathwonk]

Modmans, your thinking is right on for a cube as the mbox with top, but for the topless box, it is not a cube. A cube does have somehting to do with it though! Think about this: if you make the biggest possible topless box from 6 sq ft of material, then two of them put together at their open ends, should give the biggest possible box WITH top, made from 12 sq ft. If you believe that, what does it imply?

[modmans2ndcoming]

Modmans, your thinking is right on for a cube as the mbox with top, but for the topless box, it is not a cube. A cube does have somehting to do with it though! Think about this: if you make the biggest possible topless box from 6 sq ft of material, then two of them put together at their open ends, should give the biggest possible box WITH top, made from 12 sq ft. If you believe that, what does it imply?

the only thing coming to mind is that it should scale down so you get a similar effect with 2 boxes made from 3 sq. ft.

[mathwonk]

well the thing it implies to me is that putting the two together should make a cube of area 123 sq ft, i.e. it should give the answer to the closed top problem. So the answer to the open top problem should be the result of cutting a cube of total area 12 sq ft in half, so the answer will have base sides twice its height. The base will then have side of length sqrt(2), and height half that. so the base will have area 2 and the 4 sides will each be rectangles of area 1.

what do you think? does that make sense? try it using calculus and see what happens.

[mathwonk]

By the way, in the spirit of helping people learn calculus, we have often heard an appeal for good books. Hence also in the spirit of providing a list of answers to frequently asked questions, undertaken by Matt Grime and others elsewhere, we might compile a list of highly recommended calculus books.

These should probably be sorted and commented on by a moderator here, or other teacher, to make clear which ones are user friendly but low level books preferred by non math types, and which ones are genuinely deep treatments, for those desiring to get to the heart of the subject.

It might be useful to replace the word "good" by more descriptive ones, such as: "rigourous", "example oriented", "brief", "guided problem solving", or other terms.

Under "rigorous, authoritative and masterful", I will repeat the names of the calculus authors Apostol, Courant, Courant and John, and Spivak. There was also a terrific book by Joseph Kitchen, long out of print. All but the last of these authors have written books covering both one variable and several variable calculus. For a quick, rigorous, modern, introduction to the essentials of several variable calculus, Spivak's Calculus on Manifolds, at 140 pages, is probably unmatched. For a longer treatment, with Lebesgue integration, Wendell Fleming's Functions of Several Variables is excellent. A nice feature of that book was a 20 or 30 page summary in the appendix of an honors level introduction to one variable calculus, for those lacking the appropriate first course. Used copies for as little as $10 are available on the well known site abebooks.com

Books like those of Sylvanus P. Thompson, and Gootman, probably belong under the category "limited in scope, but highly user - friendly".

Stewart 2nd edition, and Thomas and Finney 9th edition, are thorough, standard, clear, well laid out, not overly theoretical.

The Schaum's Outline series books have been thought useful for years, but I suggest getting as old an edition as possible.

Books with titles like calculus for nitwits, should perhaps be taken at their word and avoided.

For todays web priented stduents, there are also excellent free books and tutorial sites online, like http://www.karlscalculus.org/calculus.html#toc.

[Gokul43201]

I did high school and college in India. Just to provide a frame of reference, here's a couple of randomly selected problem from the math book I used during my last 2 years of high school :
(1) (VECTORS)Prove using vector methods, that cos(A+B) = cosAcosB - sinAsinB
(2) (COMPLEX NUMBERS) Find the other vertices of a regular polygon of n sides, whose center is at z0 and one vertex at z1.

As a Physics Grad student in the US, I was TA for an introductory physics course. That was my first, and only, culture shock. There were college students that were adding fractions by adding numerators as well as the denominators !

Later I found that the high school syllabus for Physics entirely avoids using the word (and the concept of) "voltage", because it is too abstract for someone in high school. I'm not sure if this is only for this state (Ohio), but I was shocked (no pun intended) !

I strongly believe that there's not enough math and science being taught in schools in the US.

[modmans2ndcoming]

in Michigan, at least when I was in high school taking physics (1997-98) (yes I am still in college, but that has more to do with my kids and my wife's work schedule than being lazy) we certainly talked about voltage and amps. I do not see how they are to abstract for students in high school... I mean, just tell the students that voltage is to electricity as hight is to, well, a brick. they tend to get it.

I mean, voltage is no more abstract than electrons or protons, or forces.

[modmans2ndcoming]

well the thing it implies to me is that putting the two together should make a cube of area 123 sq ft, i.e. it should give the answer to the closed top problem. So the answer to the open top problem should be the result of cutting a cube of total area 12 sq ft in half, so the answer will have base sides twice its height. The base will then have side of length sqrt(2), and height half that. so the base will have area 2 and the 4 sides will each be rectangles of area 1.

what do you think? does that make sense? try it using calculus and see what happens.

yes that does make sense. using thought experiments makes it a very out of the box (pun intended) solution. but analytically, it is very apparent.

[Cyclovenom]

I am a student [a sophomore] at a University, and i've seen many of those students you all teachers has described. I've always been considered by many teachers ones of the few that actually care about learning, i've always studied the topics before they were taught, so i make sure i understand them well.

In my opinion the best way to learn is by teaching yourself, Teachers are just merely guides that can help you in case you didn't understand properly an idea.

[mathwonk]

I may be wrong, but I pride myself when I teach a course on trying to present more logical, or more insightful, or deeper versions of the material than are found in most books. I.e. I try to actually provide or suggest ideas that are not in the text.

Of course this is largely possible because the books we use are not the best available, hence there is plenty of room for improvement. But it is also true that in college, teachers usually know a bit more about the topic than is in most current books.

There are rare exceptions, but if your teachers cannot offer you anything beyond what is in your texts you might consider seeking better informed teachers, or taking more advanced courses. For example, if you are a calculus student, do you know that all monotone functions are Riemann integrable? Can you prove it? This simple fact was known to Newton, and is far easier to prove than the integrability of continuous functions, yet is omitted from most beginning calculus texts. On the other hand it is found in good ones like Apostol.

Did you know that if a function f is Riemann integrable on [a,b], even if not continuous everywhere, then it must be continuous "almost everywhere", and moreover that its "indefinite integral": F = integral of f from a to x, has the property that F is continuous everywhere on [a,b] (even if f is not), and F is differentiable everywhere the original function f is continuous, and that at such points F'(x) = f(x)?

This is a version of the fundamental theorem of calculus which is more precise than that in the most commonly used books. If you have not seen it you might enjoy proving it for yourself.

Did you know further that this is not sufficient information to recover the indefinite integral F from f? I.e. given an integrable function f, and a continuous function G which is differentiable wherever f is continuous, and with G'(x) = f(x) at such points, it need not be true that G(b) - G(a) equals the integral of f from a to b?

Can you think of a function F which is continuous everywhere on [0,1], with derivative zero almost everywhere (i.e. on a collection of disjoint sub intervals of [0,1] with total length 1), and yet with F(1) - F(0) = 1? Such a function does not obey the mean value theorem (F(1)-F(0) does not equal the value of the derivative F'(x) anywhere in [a,b]), and F cannot be an indefinite integral.

There is however a stronger version of continuity satisfied by indefinite integrals, stronger even than uniform continuity, which does suffice for this purpose. Can you discover it? If you can do any of these things without having seen them in books or courses, you are well on your way to bering a mathematician.

If you are more advanced than this already, I apologize for these elementary challenges. I could not resist trying to provide soemthing that may not have been contained in your calculus course. Even if you are already at the graduate level in mathematics, as some sophomores are, there are people here who can suggest topics of interest to you.

[modmans2ndcoming]

Mathwonk,

how about the lack of any formal Proof at all in many calc curriculums.

[mathwonk]

that is a big mistake in my opinion. formal and informal proof are the strongest features of mathematical science. everyone benefits from learning this, and so I try to include it in all courses i teach.

here is my blurb for my students:

One of the main benefits of a mathematics course is in learning to make logical arguments. (This can actually help you in arguing with a judge, or the IRS, or your boss, for example.) This means knowing why the procedures you have memorized actually work, and it means understanding the ideas of the course well enough to be able to adapt them to solve problems which we may not have explicitly treated in the lectures.

[matt grime]

If only all calc teachers took that attitude. I know one who refused to allow a question to appear on the final (multiple teachers for the section) because they hadn't taught one exactly like it. Some one asked if they'd taught how to do the preceding two questions, since the third was just doing those two questions sequentially. they had, but still refused to allow it on the final. that person won lots of teaching awards (based upon student evaluations).

[HallsofIvy]

I am a student [a sophomore] at a University, and i've seen many of those students you all teachers has described. I've always been considered by many teachers ones of the few that actually care about learning, i've always studied the topics before they were taught, so i make sure i understand them well.

In my opinion the best way to learn is by teaching yourself, Teachers are just merely guides that can help you in case you didn't understand properly an idea.

Exactly. The most important thing you can learn in school is how to learn by yourself. You won't always be in school, but, hopefully, you will always be learning.

[mbisCool]

Im currently in calculus and the entire course emphasizes computation. From what i have read this is quite common. Students are not developing thorough understandings of calculus concepts. I notice people do things such as plug and chug whatever rule they just learned when if they just stopped and looked for a second they could easily simplify the expression and the answer becomes trivial. In my class personally, the entire capter on epsilon delta proofs was skipped over with mention. I recently got a copy of spivak and immediately began working the problems. They are so much better at testing your understanding of the concepts than simple computation problems over and over. Even if the answer is derived from simple computation the exercise will be how to approach the problem not the answer itself.

Because of this i think students should work out problems that are not merely computation. Who cares if the students can calculate problems 3-32 if they dont understand whats going on or why they are doing what they are. Problems that emphasize understanding and not computation skills. Although from my experience most students at least in highschoo/cc think of math as computation and nothing else.

[mathwonk]

have you studied taylor series? have you ever noticed that most books omit much mention or detailed discussion of the series for tan(x)? It turns out that knowing the coefficients of this series is equivalent to knowing the sequence of Bernoulli numbers.

these numbers are extremely interesting, as they determine the values of the riemann zeta function at the even integers (in a formula due to euler), the order of the image of the "J homomorphism" in the theory of homotopy groups of spheres, the number of diffeomorphism classes of exotic spheres of dimensions 4k-1 which bound parallelizable manifolds, the criterion for a prime number to be "regular" in the sense of kummer, who proved fermat's last theorem for those primes, and last but not least, they determine the todd polynomials which arise in the statement of hirzebruch's general riemann roch theorem!

now why would this taylor series normally be passed over in silence, since it seems to be by far the most interesting one?

[montoyas7940]

Here is a recommendation of a good cheap, short, paperback calc book, the one by Elliot Gootman, selling new for about $15.

Thank you, I just ordered a copy. It was $4.00 used on Amazon.

[ice109]

you're fighting a losing battle. these kids come in from high school with no experience thinking about mathematics, only memorizing. you're only going to frustrate people by forcing them to do otherwise. you want to improve the state of affairs? disallow students from testing into classes above algebra and then teach those classes in your university in a challenging fashion. they will have enough facility with the formal procedures in those classes to not be frustrated by creative problems. by the time they're done with the algebra/trig sequence taught in this way their memorization habbits will be broken and they'll be comfortable with a challenging calc course.

[john16O]

To the OP, I finished calculus in the fall of my freshman year(2007). I noticed one thing, the people who wanted to do well in the class did well and the others just did what they had to do to pass, and some just failed. I understand what you are trying to do, sometimes you just have to let the people who fail, fail. I mean they are in college now. If they still have yet to take education seriously then they are in serious trouble and there is probably nothing you can do. All i can say is help those who WANT the help and let the other fall where they may...You are a very nice individual though. I would have loved to have you as my Calc teacher!

[john16O]

you're fighting a losing battle. these kids come in from high school with no experience thinking about mathematics, only memorizing. you're only going to frustrate people by forcing them to do otherwise. you want to improve the state of affairs? disallow students from testing into classes above algebra and then teach those classes in your university in a challenging fashion. they will have enough facility with the formal procedures in those classes to not be frustrated by creative problems. by the time they're done with the algebra/trig sequence taught in this way their memorization habbits will be broken and they'll be comfortable with a challenging calc course.

good point, IMO that is the problem with the schools in america. They do not teach the students how to become critical thinker/analytical thinkers. They only teach and test them on how well you memorize the material. Luckily I had a very strick father who would sit down every night and go over countless problems with me...I hated him back then but now I realized why he was doing it..

[phyzmatix]

I've scanned through most of the posts here and it seems that the general idea is that the teachers/lecturers should do something different or that crap students might only be crap because of something their teachers are doing wrong and not because they might just not have what it takes. Where I agree that a teacher can obviously make a difference, I personally believe that the problem lies with students mostly and their general lack of maturity when it comes to their approach to life (and studying).

Honestly, with the exception of a few, how many 19 year olds (well, I don't know how old they are when they hit university in the states, but over here that is the average age) know what they want from life and/or have learnt the value of an education where your ultimate goal is the accumulation of knowledge and not just the receipt of a piece of paper?

Mathwonk mentioned something about working in a factory and I had a similar experience. Totally screwed up my first attempt at Uni and after seven years of menial jobs and mind-numbing employment as an unskilled worker I just had enough...

This time around, a lot of conscious thought went into my choice of degree as opposed to the first time when it was pretty much a coin toss and "seemed like a good idea at the time". Some things are more difficult being a student now than it was when I was younger, but some are easier, e.g. the motivation, determination and the understanding of what and why I'm doing what I'm doing is on a different level than before.

Oh yes, and of course the predominant idea of what it means to be a student no longer involves alcohol, exotic substances and finding someone to play with mini-me... :tongue:

[mbisCool]

I remember when my class was first learning limits in calc. I wasnt sure how to calculate the limit of sin(x) / x, as x goes to 0. When i asked my teacher how to approach this problem he told me "the limit is 1, its just something you memorize."

I then began to question his expectations of the class.

[Moonbear]

Where I agree that a teacher can obviously make a difference, I personally believe that the problem lies with students mostly and their general lack of maturity when it comes to their approach to life (and studying).

Honestly, with the exception of a few, how many 19 year olds (well, I don't know how old they are when they hit university in the states, but over here that is the average age) know what they want from life and/or have learnt the value of an education where your ultimate goal is the accumulation of knowledge and not just the receipt of a piece of paper?

That is part of the overall picture of things educators need to consider. The maturity of students hasn't drastically changed, at least not in the time I've been teaching. Yes, I see slight variations from class to class in how seriously they take their studying, but overall, this doesn't change much. So, if someone teaching an undergraduate course expects their students to have the same level of sophistication and ability to work independently as students in a graduate level or professional program, they're not doing the best they can to teach those students.

Likewise, not everyone has the same learning style. I think because different teaching approaches in some fields have self-selected those learning styles among the educators (the ones who learned well with the old teaching methods will be the ones who move on and eventually teach themselves), it takes some real mental stretching on the part of the educators to address different learning styles and ensure part of the class isn't left out simply based on their predominant learning style.

[physics girl phd]

The maturity of students hasn't drastically changed, at least not in the time I've been teaching.

My questions: Is there a change in the preparation/preexisting skills of students?
I have no data/insight here. I see prepared students and unprepared students.

Is there a change in the way they approach classes?
I again have no data here... but via a conversation with my husband yesterday: when I was a student, I considered attending classes and doing all the assigned problems "my job", and I also thought it impolite to skip classes, regardless of the skills of the instructor. I also did all problems the professors recommended, checked my results with the professors hand-writtten taped-on-the-wall solutions and went to office hours if I didn't understand them, even though I wasn't given credit for them towards my grade (a few professors would collect problem notebooks and give a bit of extra-credit at the end of the term if you needed a slight curve to get to the next grade).

I went to a private undergraduate institution, while I now teach physics at a public university; but my husband, who went to a large land-grant university for his undergraduate degree also said he would never have considered skipping classes as an undergraduate. I also note that students don't do the recommended problems and often don't even come to class unless homework and attendance are built into the grade. An undergraduate student working for me this term noted that she skips ~ 1 class per week "without even blinking" at doing so.

While I need to do a literature search, I think part of the "dirty-little secret" to clickers and some techniques that I use (like team learning) are that these techniques are ways of building attendance into the grade. Any peer instruction may be just a secondary bonus.

Maybe I'm just down-and-out today. :grumpy: Even though I announced my lecture was covering a chapter that was removed from the present edition of the text, I had 10 students (of a class of about 110) decide to leave my class yesterday since I was giving a lecture and not having a graded team-learning "in-class work" (it's difficult to do "activities" covering some thermodynamics topics). While I confess to not enjoying the process of lecturing, it was still a good lecture.. . I had tons of demos (including a big flame when I discussed and performed the process of lighting a Bunsen burner... with a large propane tank, since there is no gas hook-up in our demo hall and our "campstove" burner is broken and I need a high-heat source for my linear expansion demos). I had the students laughing when I talked about buoyancy and overfeeding my fish when I was little... increasing their mass so that they couldn't fill their air-bladders up enough to be buoyant). I could tell by looking around the room that I had the students engaged. Some of my students have complimented me on my lectures... saying they are still interesting. I know they are better than many of the lectures I attended as a student.... I can only recall ~5 demos ever being used in a lecture when I was a student (including both my physics and chemistry classes!)!

Is this a problem with internet tools?
Do these students who leave think all the information they need for a test is online on the posted slides and pre-class quizzes? In my case, it really isn't... I think I'm using the tools well. My demos and extra information (in both lectures, feedback to the quizzes, and in-class works (where I make sure to visit all groups & ask extra questions, etc.) are often a starting point for my test questions... which are VERY applied. My husband's view is that these are students that probably really shouldn't be in college... this is the "death of the university."... and even the death of our society... and we need to provide other sources of training for jobs that should be done in the US (note: we are still disturbed that the machines that mint our coins are made in Germany). Also: for math based classes... is this a problem with online homework systems? I'm still trying them out, and haven't really been pleased by what I've used yet... my students had better results when I had them turn in problems on paper and I manually graded them.

Well, this turned into a minor rant despite the fact I felt I had a few important questions to get some other viewpoints on! ... sorry... :blushing:

[Moonbear]

I really don't think students are that different. When I was in college, on the first day of class, the lecture hall was packed to capacity. After about the third week of class, about half the seats were empty. There were always students whining for extensions on assignments (I HATED that, because if they were granted, I thought it was unfair to those of us who had gotten our work done and turned in on time). Though, one difference is that the lecture halls used to be designed in a way that the lecturer could hear the students talking in the back, so could ask them to leave. I've realized that I cannot hear the students talking in the back of the classroom in the modern classroom, which means I cannot stop them when I think it is getting to a level that is disrupting other students.

Of course, when I was a student, even though I attended lectures, if a lecturer was boring, I'd often be sitting in the back working on a crossword puzzle rather than listening. I'd look up every so often to copy the next board-full of notes.

Though, do students NEED to attend lecture as often today? If we record our lectures and they are available for them to listen online, does it matter if they are sitting in the classroom if they can get the same information online? For some students, yes, seeing the non-verbal cues, facial expressions, gestures, etc., helps them to focus on what is important. But for many, they might do better just listening online. I wish I had that available when I was a student. No matter how hard I tried, about halfway through any lecture, no matter how engaging, I'd start losing focus (our lectures were 120 min long, and a double lecture was a full 3 hours...I couldn't even get through that without needing a restroom break somewhere along the way). It would have been nice to be able to listen at my own pace to the lecture online, complete with all the figures, rather than trying to decipher my notes as they drifted off to a squiggly line while I nodded off. If I missed something, there was no going back, and those old tape recorders didn't work very well.

Though, to me, 100 students in a class is a SMALL class. I was usually one of 300-500 in lecture courses.

But, on the other hand, when they show up at my office all teary-eyed because they are struggling to pass the class, and I've never seen them in lecture, I don't feel the least bit troubled by telling them there's nothing I can do for them other than to work with them on improving their study skills.

As for demos, when I was a student, the chemistry and physics courses were FILLED with demos. Well, actually, only one of the two lecturers for the chemistry course gave demos. I used to go to the lecture for the one who did demos, because it was just so much easier to learn when I was staying awake to watch demos; or, if I thought my own lecturer would notice my absence, I actually attended BOTH sections with both lecturers...that was probably the best thing I ever did, even though it doubled my time in lecture, because I got to hear the same material presented two different ways. I'm not sure I got a lot out of the demos themselves so much as it was a break from the monotony of lecture that woke me back up and regained my focus on the remainder of lecture. The biology courses didn't have much in the way of demos, but that's because they were all combined with lab courses, so we got plenty of hands-on experience without demos.

[cdotter]

When I took calculus in high school, my textbook used a lot of examples. We covered some proofs...while I don't remember any of them, some were interesting but most weren't (to me). I'd much rather do practice problems using real life examples and not only get the right answer, but understand WHY I am doing what I am doing. But as for the underlying principles of mathematics, I don't care for them.

I guess you could think that for some, calculus is just a tool. A carpenter uses a hammer, but he does not care about the underlying principles of how and why the hammer works.

[mathwonk]

i am inclined to agree with comments putting the explanation for poor performance on how young college students approach their courses. I have recently had some of the smartest, least successful classes I have ever had. Some of the students do not attend regularly nor do any of the extra work recommended to do well.

The attitude of doing only what is required or "due", seems to explain the poor performance of these very talented underachievers. So it is not only that understanding has not been expected, but that independent work has also not been expected. This may not differ from past years, but there is more pressure today to excuse it, rather than letting people fail, because there are so many who would fail.

[fluidistic]

Hi,
I'm a freshman and passed calculus I and II. Here in Argentina and more precisely in my University I think I was taught calculus from a different manner than it is usually taught in the US I think.
Theoretical part (a big classroom with about 100 people): The professor writes on the blackboard definitions, then lemmas (with their detailed proof), then theorems, etc. Everything that can be proved is proved on the blackboard so that we can take notes of the proofs of the theorems and lemmas. The lecture lasts 2 hours. After this we enter into another classroom. It's 2 hours of practice (there are about 4 helpers for 25 students). Here we buy the sheets of problems or they give it to you. Basically there are many different kind of problems and some ask you to prove relations. As there is no "Introduction to demonstrations" course, many students find this part the most difficult. But it is a very good training.
Tests are made of a theoretical part (about 30% of the test) and a practical one. To pass the test you have to pass BOTH the theoretical part and the practical one. (you have to score more than 40% on both parts in order to success the test.)
The theoretical part consists to demonstrate 2 or 3 theorems/lemmas or relations you never dealt with before. So this is clear : you cannot success it by memorization. It's impossible.
Furthermore the final exam which is the only exam that count for your grades and the only exam that can make you pass the whole course is much harder than any test. The professor gave us a list of 45 demonstrations (He gave only the name of the theorems/lemmas and not the demonstrations. We have to find them in our notes, books, etc.) that could get into the final exam. What to say about the practical part of the exam? Well it can be any kind of exercise... so you have to have dealt with all kind of problems.
And one more thing I can say : we don't use any calculator in any math course. So integrals are calculated handily and so are series and whatever you can imagine. This means you cannot check out the result you got for an integral with a calculator.

I want to add that the professor said as advice to use the book from Spivak even if not alone. He clearly told us to study hours at home everyday not to get lost with all the new stuff coming fast. (3 months to cover calculus I and 3 months to cover calculus II).
All professors including the Physics' ones too told us that University is VERY hard home working... you have to study at home by your own, checking out books and doing a lot of exercises. I think it's clear that University is hard for almost everyone and it should be told to freshmen. They can't ask "what will be tested?", they should understand the chapters covered and have done many different exercises. In one word they must be prepared and it's not the job to the professor but their job. The professor and helpers are there to help the student to success, not to make him success.

Having said that I liked calculus and the way it is taught here. You cannot pass if you don't understand the matter. That's why more than a half of the students give up during the first year.

[lurflurf]

I remember when my class was first learning limits in calc. I wasnt sure how to calculate the limit of sin(x) / x, as x goes to 0. When i asked my teacher how to approach this problem he told me "the limit is 1, its just something you memorize."

I then began to question his expectations of the class.

That is good advice. Although I might say pick the sin that makes the limit 1 because it is easier to remember.

[khemix]

i am inclined to agree with comments putting the explanation for poor performance on how young college students approach their courses. I have recently had some of the smartest, least successful classes I have ever had. Some of the students do not attend regularly nor do any of the extra work recommended to do well.

The attitude of doing only what is required or "due", seems to explain the poor performance of these very talented underachievers. So it is not only that understanding has not been expected, but that independent work has also not been expected. This may not differ from past years, but there is more pressure today to excuse it, rather than letting people fail, because there are so many who would fail.

the problem is, with math in particular, that lectures are not very useful. its not the fault of the professor really, as its difficult to learn and understand math in the mere 2-3 hours alotted a week. math is simply too abstract to be digested by a mere lecture, and if you are lost in one step, the entire lecture becomes a textbook copying section. this contrasts say biology where the lecturer can show motion pictures and diagrams to illuminate the textbook readings. this is why math class attendence is low almost everywhere.

as for the attitude of doing only what is "due", this too is very logical. there is simply too much college homework for anyone to keep up with. if people were to do all the suggested questions in all their courses, they would be running on 5 hour sleep. because all that matters is grades, most people will only focus on getting assignments in so that they can succeed.

i personally think every math prof should write his own textbook for his class to use, and that this should be the students main source of information. additional examples could then be provided in lecture. i mean seriously, is there any reason to be copying lemmas and proofs word for word on the board?

[mathwonk]

one thing that can help is to actually listen and ask questions during the lecture. also if you pay attention and know what is coming up you can read up on it in advance, and then it is easier to follow the proof. it may seem there is no time to study this well, but in fact there is. and in fact it is the only way to learn the material well. i have often written a textbook for my classes but this does not prevent some students from refusing to read it. declining to ask questions, or come to office hours, or even to class. if time is important then never ever miss class. a professor can explain in class material that will take three or four times as long to learn independently, so every hour of class missed is 3 or 4 hours of time lost. and often it is simply impossible to recover what the professor gave.

[mathwonk]

let me give another example of how illogical it is to only do what is "due". on a test, if there are 4 problems required and one "extra" problem, some students will not even attempt the extra problem. but if there are 5 problems, and none is called extra, most will attempt them all. where is the logic in that? no matter what the problem is called it is still worth points.

but some students will simply not do anything they do not think they have to. so if i want to increase my students scores, instead of having 4 required and one extra problem, if i simply give 5 problems and score the test over 80 instead of 100, they will do better because they will try them all.

[khemix]

reading in advance defeats the purpose of going to lecture. lectures are supposed to give one a grounding in what will be read. this works in less abstract courses, but in math i personally come out with very little from a math class expect some historical facts.

i dont know if its a matter of intelligence, but very few people can keep up with a math lecture no matter how hard they listen. as you are reasoning one step, the prof is already on another, so either you be stubborn and try it out or not risk missing the new information. i sit there and day dream and take notes, because at this point the effort isnt worth it. i know ill get a real understanding when i read the book. because the book is patient. i cant stop and ask the prof hundreds of questions because its unfair to the class, and he has an agenda to go through. this is why even going to class is useless; the course is in the textbook.

you are correct in saying students will only do what is required. efficiency is key in undergrad, and you rarely have time to go beyond the requirements because there is likely another course that needs your time. also, any free time you can snatch isn't likely to be spent on more work - a break is very nice from time to time.

as for the extra problems on tests, i can only give my personal student prespective. if a problem is listed as extra, i immediately assume its super hard and is only attempted by the best students. this and extra problems tend to have stricer marking schemes.

[Moonbear]

the problem is, with math in particular, that lectures are not very useful. its not the fault of the professor really, as its difficult to learn and understand math in the mere 2-3 hours alotted a week. math is simply too abstract to be digested by a mere lecture, and if you are lost in one step, the entire lecture becomes a textbook copying section. this contrasts say biology where the lecturer can show motion pictures and diagrams to illuminate the textbook readings. this is why math class attendence is low almost everywhere.
Except students have the same attitude in biology classes. What the students don't seem to recognize is that if they get lost during lecture, they can make a note about where and now they can save time reading their textbook by going straight to that section rather than having to read the whole book end-to-end with no idea of what is important.

as for the attitude of doing only what is "due", this too is very logical. there is simply too much college homework for anyone to keep up with. if people were to do all the suggested questions in all their courses, they would be running on 5 hour sleep. because all that matters is grades, most people will only focus on getting assignments in so that they can succeed.
The workload in college today is lighter than when I was a student, and even lighter yet than when some of the older faculty were students. I have access to years of course materials and exams from the now deceased professor who used to run the course I'm taking over, and it's amazing how much has been cut out over the years. Yet, students used to be expected to learn it all...and did. College IS full time study. It doesn't help that students all seem to think the weekend starts on Thursday night now. Cutting out a day of studying certainly makes it harder to get everything done, but that is not the fault of those of us who are teaching them.

It's frustrating that students don't take any responsibility for their own learning. If I don't have mandatory attendance, or give quizzes in class, or require something be turned in every class period, they don't show up to lecture or try to cram the week before the exam rather than keeping up with the content all along.

i personally think every math prof should write his own textbook for his class to use, and that this should be the students main source of information. additional examples could then be provided in lecture. i mean seriously, is there any reason to be copying lemmas and proofs word for word on the board?

Do you have any idea how long it takes to write a textbook? Even more so, how long it takes to write a GOOD textbook? Why reinvent the wheel if there are good textbooks already available?

As for why to use examples directly from the book to write on the board, while this isn't my favorite practice, it sometimes makes sense. For example, to make sure all the logic of each step is explained clearly. And, when you complain about how much work there is, this is cutting down on the amount of reading you need to do outside class so you can spend the time working on practice problems. I had a student this week ask me what to read in their textbook (not math) because "there's a lot in the book" on the current topic. We spent 4 lectures on a single chapter. I pointed out that I covered more in my lectures than is in the book. If anything, the book I'm currently stuck with (will change next year...it was ordered this year before I was hired to teach the class) is very inadequate for the material covered and glosses over important topics, so I went into much more detail in my lectures.

Part of the problem I'm seeing, though, is that large numbers of students have already made it into their sophomore year with atrocious study skills. When I taught freshmen, I expected this...they all came from different backgrounds, and some could breeze through high school without studying. But, that should be caught in freshman year and corrected. Now that I'm teaching sophomores, I'm still seeing a lot of students who basically waste a ton of time studying because they aren't doing it at all effectively.

[mathwonk]

khemix, at least you are posting your views here, which helps the discussion, but some of your views are sadly incorrect, and they are holding you back. the idea that trying to learn enough by reading in advance to be able to actually follow a lecture you yourself think is almost impossible to follow cold, is ludicrously self contradictory. you are clearly intelligent, but opinions like that one, and your argument for not trying extra problems, sound as if you are trying to give yourself an excuse to be lazy.

[Moonbear]

reading in advance defeats the purpose of going to lecture. lectures are supposed to give one a grounding in what will be read. this works in less abstract courses, but in math i personally come out with very little from a math class expect some historical facts.
My advice to my students is to SKIM the textbook chapters ahead of the lecture. Know the overall content and direction of the assigned reading before walking into lecture, and maybe even have a few questions in mind of content that didn't make sense the first time through. That will help follow the lecture more easily, and ensure you know the right place to stop and ask questions if the areas you KNOW the book is not going to adequately cover for you are also not adequately covered in lecture. After the lecture, go back and reread the sections that were emphasized for detail and to ensure your lecture notes make sense and are complete.

i dont know if its a matter of intelligence, but very few people can keep up with a math lecture no matter how hard they listen. as you are reasoning one step, the prof is already on another, so either you be stubborn and try it out or not risk missing the new information.
If you spent some time looking over the book BEFORE lecture, you would have an easier time following the lecture. This is the problem of not even looking at the chapter before class. If you have already read the general content and know where the tricky parts are, you can switch your focus in class from just trying to keep up with taking notes to listening for tips and hints on how to get through the sticky parts. And, you know the right places to ask questions.

You seem to think professors got where they are without having been students too. We know what the undergraduate course load is like, and that it can be done.

i cant stop and ask the prof hundreds of questions because its unfair to the class, and he has an agenda to go through.
No, you can't ask hundreds of questions, but you can ask a few if he went over a point too quickly and everyone else is just as confused about it.

you are correct in saying students will only do what is required. efficiency is key in undergrad, and you rarely have time to go beyond the requirements because there is likely another course that needs your time. also, any free time you can snatch isn't likely to be spent on more work - a break is very nice from time to time.
As in my previous reply, the problem is that students aren't very efficient. They tend more toward hastiness than efficiency. If you are really efficient in your learning and studying process, it does not take as much time, and you would have time to do the extra work. In fact, that extra work IS necessary for learning and part of efficient studying. Just because you aren't told to hand in a problem set for a grade doesn't mean that you don't need to practice those problems to do well on an exam.

as for the extra problems on tests, i can only give my personal student prespective. if a problem is listed as extra, i immediately assume its super hard and is only attempted by the best students. this and extra problems tend to have stricer marking schemes.
Okay, so they're going to be hard and graded harder...why is that a reason to not even bother attempting it? If you've struggled with the entire rest of the exam and run out of time for that problem, then it makes sense to skip that problem, but just because it might be hard is no reason to skip it. That's just laziness, and is precisely the problem seen in students today...they aren't interested in challenging themselves to learn as much as possible, only as little as is required. If you have an hour for an exam, and there are still 10 min left at the end of finishing the required problems, why would someone choose to hand in the exam early rather than sit for the remaining 10 min and see if you can get a couple extra points on the challenge problem? When I was a student, we knew those problems were hard, so we didn't get frustrated if we couldn't get the solution, but there was no harm in giving it a try...in fact, we could only benefit from getting a bonus point or two even if we couldn't get the whole problem solved. It could make up for a silly mistake somewhere else in a required problem, or give an extra cushion if you get a low score on another harder exam, or a bad quiz grade on a day when you're not feeling well.

I think part of the problem is students take university for granted rather than recognizing it as opportunity. You have 4 years when your only requirement or responsibility is to learn as much as you possibly can to prepare you for any number of paths when you are done. Never again will life be so easy.

Then again, if a student really only cares to do the bare minimum to pass a class, it is their choice. They will receive the grade that reflects they did the bare minimum to pass, which in my courses is a C. If they want to be C students, I can't force them to do better. However, they also shouldn't be the ones showing up at the end of the year complaining that they didn't get a B.

[epenguin]

One thing I think does work, is the patience shown by people on this forum, at helping people without doing their work for them.

I am losing it. :grumpy:
The patience I mean.
Mainly caused by the about half who post a question one responds to, and then never come back.
But I saw yesterday a post by someone who came back after a year and said he had solved it so maybe I am not patient enough! :biggrin:

[cdotter]

If you spent some time looking over the book BEFORE lecture, you would have an easier time following the lecture. This is the problem of not even looking at the chapter before class. If you have already read the general content and know where the tricky parts are, you can switch your focus in class from just trying to keep up with taking notes to listening for tips and hints on how to get through the sticky parts. And, you know the right places to ask questions.

This isn't directed towards you, but I really hate this kind of reasoning. I am paying tuition to be taught, and not have a professor regurgitate a textbook.

[mathwonk]

cdotter, it is hard even to understand your complaint. the point is that any significant math subject is really hard and has many ramifications, not all of which are ever present in any one particular discussion. Even if you read everything in the assigned textbook you will likely not completely understand it, nor even encounter much of what the professor will say in class. what you will do is prepare your mind to better understand what is said in class.

If indeed you do not want the prof to merely regurgitate what is said in the book, you are even more recommended to read it yourself in advance, to contrast that presentation with the professor's. and if you read the book in advance you just might come up with a question you want to ask.

after reading your post three times, i think your mistake is assuming that moonbear was saying the lecturer is merely going to regurgitate the book to you. the subject itself is much larger than what is said in the book and the professor hopefully has a global grasp of it.

only with a particularly dull class, or one which does not itself ever read the book, does the average professor ever limit herself to saying only what is found there. still some classes complain if the class discussion contains more than the book's does. some students think a course should "cover" the content of one assigned book, and not go beyond.

Actually the idea is to convey a good impression of the nature of the topic, and the book is merely one resource. There is no rule that students may not go to the library and read even more books on the same topic. If they do and they choose a good one, it will barely seem to resemble the one for the course, as only monkey see monkey do books all treat the subject the same way.

[khemix]

Except students have the same attitude in biology classes. What the students don't seem to recognize is that if they get lost during lecture, they can make a note about where and now they can save time reading their textbook by going straight to that section rather than having to read the whole book end-to-end with no idea of what is important.
I think its different in biology. If you are unclear about one section in a biology lecture, usually the following sections you can still understand because the subject is intuitive. In math, everything builds up from previous results. Whats worse, is you won't understand any math until you have all prior results down cold. With science courses, you can still sort of ignore past results.

Do you have any idea how long it takes to write a textbook? Even more so, how long it takes to write a GOOD textbook? Why reinvent the wheel if there are good textbooks already available?
I imagine no longer than writing them on the board. The book doesn't need to be a classic with 50 questions in each chapter. But this problem only arises in courses where the teacher does not follow the book closely. In such a case, the teacher should seriously consider adopting a new book. Or atleast type up lecture notes. Is it really neccessary to be copying theorems? And at the same time be listening...

[morphism]

I imagine no longer than writing them on the board. The book doesn't need to be a classic with 50 questions in each chapter. But this problem only arises in courses where the teacher does not follow the book closely. In such a case, the teacher should seriously consider adopting a new book. Or atleast type up lecture notes. Is it really neccessary to be copying theorems? And at the same time be listening...
You "imagine"? Clearly you've never tried this, then. Moreover, it's not like professors simply wing their lectures all the time either; most actually prepare a set of notes beforehand. And have you tried to TeX anything? It's not exactly a pleasurable experience. So if a professor chooses not to waste his/her time typing up lecture notes, I will not hold that against him/her. These people have better things to do.

Your posts in this thread really show a great level of ignorance. But this doesn't come as a surprise to me, having read an earlier post of yours on people who pursue PhDs (http://www.physicsforums.com/showthread.php?t=267151#12).

[khemix]

You "imagine"? Clearly you've never tried this, then. Moreover, it's not like professors simply wing their lectures all the time either; most actually prepare a set of notes beforehand. And have you tried to TeX anything? It's not exactly a pleasurable experience. So if a professor chooses not to waste his/her time typing up lecture notes, I will not hold that against him/her. These people have better things to do.

Your posts in this thread really show a great level of ignorance. But this doesn't come as a surprise to me, having read an earlier post of yours on people who pursue PhDs (http://www.physicsforums.com/showthread.php?t=267151#12).

So scan what you write and compile it into a notebook students can buy. I've used math symbols, and I agree they are time consuming. Still, if I can do 4 page lab reports every week full of them, a book should not be that much of a problem. My point is math classes should focus more on learning instead of writing out the textbook, a view that many students share. This can be done by preparing lectures in advance, the way science courses do.

How do my posts show a level of ignorance? I am sharing my own views that are shared by many undergrad students. I don't pretend to know the master plan behind undergrad work or deny professors know what is best for me. I am just saying that math lectures at their current state are not very useful.

[And I still hold my view on PhDs, despite the fact that it is completely off topic. People persue them for fame, fun, or money; not for the 'betterment of humanity'.]

[mbisCool]

[QUOTE=khemix;1981647]So scan what you write and compile it into a notebook students can buy. I've used math symbols, and I agree they are time consuming. Still, if I can do 4 page lab reports every week full of them, a book should not be that much of a problem. My point is math classes should focus more on learning instead of writing out the textbook, a view that many students share. This can be done by preparing lectures in advance, the way science courses do.
QUOTE]

Personally, I prefer when my professors lecture as we get to cover more material in this manner. Any lectures I have gotten in a math or science courses were far from verbatim copies of the text. They offer another unique perspective to the material which increases clarity. The professor generally knows the material well and emphasizes the important or difficult topics

[TMFKAN64]

if I can do 4 page lab reports every week full of them, a book should not be that much of a problem.

Congratulations! We have a winner!! This is without a doubt the stupidest thing I've ever read on PF! Clearly there is no real difference between a 4 page undergrad lab report and a 300 page textbook!

[Moonbear]

This isn't directed towards you, but I really hate this kind of reasoning. I am paying tuition to be taught, and not have a professor regurgitate a textbook.

And if all my students read their book before showing up to lecture, it would be completely unnecessary to repeat the content in the book when giving the lecture. Since the vast majority take khemix's lazy way out, they cannot follow the lecture if we don't repeat much of what's in the book.

I think its different in biology. If you are unclear about one section in a biology lecture, usually the following sections you can still understand because the subject is intuitive. In math, everything builds up from previous results. Whats worse, is you won't understand any math until you have all prior results down cold. With science courses, you can still sort of ignore past results.
You've clearly not taken many biology courses, and certainly not mine. Everything builds on previous subjects. Of course, the BAD students don't see this, because they are missing these concepts, but the astute student recognizes it.

I imagine no longer than writing them on the board. The book doesn't need to be a classic with 50 questions in each chapter. But this problem only arises in courses where the teacher does not follow the book closely. In such a case, the teacher should seriously consider adopting a new book. Or atleast type up lecture notes. Is it really neccessary to be copying theorems? And at the same time be listening...

:rolleyes: I'm going to say this bluntly. You haven't a clue about how classes are taught or how textbooks are written. Go back and reread the rest of the thread and try to learn something.

You should also look at the contradictions in your own arguments. You complain if a lecturer just repeats the book, then complain if they don't follow the book closely. Which is it? Ideally, lecture and the textbook should COMPLEMENT each other, not be verbatim copies of one another.

[mathwonk]

khemix, please go to my website and take a look at a tiny fraction of the notes i have written and provided free for students to use. then see how long it takes you to read some of them. then after you have done so, get back to me.

[khemix]

[QUOTE=khemix;1981647]
Personally, I prefer when my professors lecture as we get to cover more material in this manner. Any lectures I have gotten in a math or science courses were far from verbatim copies of the text. They offer another unique perspective to the material which increases clarity. The professor generally knows the material well and emphasizes the important or difficult topics

As do I, which is why I suggest they type their own notes up and teach from that. Profs usually have shorter, more elegent proofs than found in the book. Also, if we use notes exclusively written by them, we are more clear of the profs and the particular schools expectations.

Congratulations! We have a winner!! This is without a doubt the stupidest thing I've ever read on PF! Clearly there is no real difference between a 4 page undergrad lab report and a 300 page textbook!

For Pete's sake, I'm not suggesting typing up a hardcover text written for use world wide. There is usually 2-3 professors that teach a course. They can easily type up a 300 page course text similar to all the lab manuals we have printed every year. Lab co-ordinators are often writing new ones each year, and they span 200-500 pages. Don't need all the publishing rights things, keep it local. Or at the very least scan the notes online. DO profs have to go through all this when they write manuals or supplementary notes? Lab co-ordinators pull off just fine...

Typing with mathfont is not very fun, but its not as difficult as some make it out.

You've clearly not taken many biology courses, and certainly not mine. Everything builds on previous subjects. Of course, the BAD students don't see this, because they are missing these concepts, but the astute student recognizes it.
I have taken 1.5 full bio course, and am enrolled in 1.5 now. Yes, everything builds on previous results; this is true for any course. My point is its easier to swing by in a course like bio or chem when you've missed a lecture or two. You can still learn about a new biochemical reaction even if youve missed the old one. With math, its more difficult because everything you do builds on last weeks material, unless you start a completely new topic.

khemix, please go to my website and take a look at a tiny fraction of the notes i have written and provided free for students to use. then see how long it takes you to read some of them. then after you have done so, get back to me.

I have. I found the elementary algebra ones good as they had detailed explanations. The linear algebra ones were way over my head, they were too terse and didnt explain a lot of the terminology (abelian group??). However, I'm sure in a lecture of yours you would elaborate on this or your students would have the proper background.

You should also look at the contradictions in your own arguments. You complain if a lecturer just repeats the book, then complain if they don't follow the book closely. Which is it? Ideally, lecture and the textbook should COMPLEMENT each other, not be verbatim copies of one another.
I don't mind the lecturer repeating the book, but why bother writing our all the theorems on the board? Why not just talk about it (ie. turn to page 105, here we have Sard's lemma, why don't I give some more examples and see why the hypothesis is true). If they don't follow the book, then the text is useless so we have a very crappy source to learn it from - rushed notes on the board w/o any explanation. If you expect students to find the time to read two entirely different spins on the same topic, regardless of whether or not they are insightful, I am not surprised you students don't live up to your expectations.


I dont know why I'm being attacked on this thread. I already said I dont pretend to be the master of any subject or the art of learning. I just thought I'd share some what me and many of my classmates believe. If you'd rather sit there and believe we're all lazy and unmotivated be my guest. The fact is I know a large group of students who love our subject matter, but it is hard to be motivated with all the pressure. Some profs just expect too much. Real learning does not result in the best grade. Because grades are what get me into programs, I am afraid I will continue to learn the course minimum to do well on tests and exams.

[CoCoA]

which is why I suggest they type their own notes up and teach from that.
Actually, you suggested that they supply those notes to students, not just teach out of them. The level of polishing of the notes for this endeavor is what will take so much time, not just simply writing them. That's an unreasonable expectation.

I don't mind the lecturer repeating the book, but why bother writing our all the theorems on the board? Why not just talk about it (ie. turn to page 105, here we have Sard's lemma, why don't I give some more examples and see why the hypothesis is true).
I agree with this, at least to some extent. I think referring to the text instead of writing everything out implies that the lecturer has referred to the text also, and supplies an example for the student to do so.

On the other hand, not writing things on the board can easily be taken too far. Thinking about mathematical ideas takes time. If a professor talks too much about several things in a row from the book without writing down enough stuff on the board, the professor will just talk faster than students can absorb. In a sense, the most important part of writing is to control the pace of the lecture.

The fact is I know a large group of students who love our subject matter, but it is hard to be motivated with all the pressure.
Anyone that loves their subject matter will do the extra exercises in homework, the extra problems on the test, etc. Claiming love for a subject and not doing that stuff is like claiming love for a girlfriend and then blowing her off.

if there are 4 problems required and one "extra" problem, some students will not even attempt the extra problem.
funny, this was a talk at the last joint meeting:
Tristan Denley, "Students Don’t Do Optional."

[mbisCool]

I agree with this, at least to some extent. I think referring to the text instead of writing everything out implies that the lecturer has referred to the text also, and supplies an example for the student to do so.

On the other hand, not writing things on the board can easily be taken too far. Thinking about mathematical ideas takes time. If a professor talks too much about several things in a row from the book without writing down enough stuff on the board, the professor will just talk faster than students can absorb. In a sense, the most important part of writing is to control the pace of the lecture.

Ideally the lecture would compliment the text and the text compliment the lectures. I personally feel the act of writing my notes helps me remember them very well. I assume this and the reference they provide are why lectures are presented on the board...

[mathwonk]

khemix you inspired some heat because you complained that your professors were being paid to teach you, but when we also tried to teach you something you apparently rejected it, revealing that you are not so easy to teach. you are at least partly your own problem and you seem to be blaming someone else. you remind us of why it is so hard to help some people. but we piled on a bit. my apologies.

and if you look at my notes for math 8000, abelian groups are defined on page 1.

even in the math 4050 notes the qords are used but no knowledge of them is required. dont be afraid of words which are explained in context in the sentences where they appear.

those notes state explicitly that the abelian groups being discussed are products of Z/n 's, which presumably all elementary algebra students have seen.

[Moonbear]

Some students also just learn better by listening than through reading, so lecture serves them far better than a textbook ever will. And, as we all know from the various misunderstandings that occur on forums such as this, speaking to someone carries a lot of non-verbal information that helps convey context, importance, relevance, and understanding in a two-way direction between both lecturers and students that cannot be accomplished in any written text.

I give my students my notes...I say a lot more than is written. I had one comment this week that there is a LOT in the textbook, and what should she focus on while studying (this is a student who actually attends lecture). I was a bit taken aback, because my impression is that the book is incredibly skimpy on the subjects we were covering in this unit. I suggested focusing on my lecture content, because there was a lot more covered in lecture than in the book. They don't realize that my hour of talking covered more material than 3 hours of them reading.

Of course, reconsidering what khemix has been saying, perhaps he just has bad experiences with bad lecturers. We do know they exist, the lecturers who just practically read the book to the students and provide no additional information or insight, and don't take time to help them connect concepts or make associations between related materials. Perhaps that is khemix's only experiences. Of course, that is not the intention of this thread to complain about the bad lecturers, but to focus those interested in improving their teaching on the GOOD lecture techniques.

So, I'm going to ask that we get this thread back on topic. I really don't want to lock it when there is possibly still valuable information to discuss, but non-constructive complaints are not going to keep this thread going.

[buffordboy23]

The biggest task I have seems to be helping students learn how to learn. Some fail to come to class, others never look at the notes they take, and many seem not to even open the book.

Many never ask questions, and those who do, often ask things that could be found immediately by looking them up in the index of the book. People who ignore office hours for weeks expect me to schedule extra help sessions the day before the test. Questions more often focus on "what will be tested?" instead of how to understand what has been taught.

Everyone seems to have taken calculus in high school, but most also seem not to know anything about algebra or geometry or trigonometry. With the advent of calculators some also do not know simple arithmetic, like how to multiply two digit numerals. (I have had students who had to add up a column of thirteen 65's on a test, apparently not knowing how to multiply 13 by 65.)

Many think that having taken a subject "2 years ago" is a valid excuse to have forgotten the material, and to expect the teacher to reteach the prerequisites. Appparently no one ever dreams of reviewing the prerequisites before the course starts. Books like "Calculus for cretins" are apparently more popular than books like "Calculus for science majors".

When I was in college students like this were just ignored, or expected to flunk out, but in today's setting there are so many like this that they form the primary market. With all good faith to teach these stduents, the failure rate is still about 50% in college calculus across the nation, in my opinion. What are some ideas on how to improve this?

I realize that this is an old post, but I saw that it has been resurrected. I am an ex-teacher who returned to college to study physics, so I have recent experience regarding both educational windows, as a student and as a teacher.

I see what you are saying in my upper level physics courses. Basically, many students don't perform well due to many reasons, so grades are curved and a large majority of students pass the course although maybe a significant number of them shouldn't. What is the overall benefit to the mathematics or physics student for such behavior from an academic institution? Because they get away it, these students do enough to get by, but not enough to gather a deep understanding of the material and to develop a serious discipline in regards to their learning. So, the student never matures. Eventually, they will graduate with a degree, but they are not adequately prepared for graduate school or to enter into their job field.

If I were a professor, nothing would be more easy than to take the hard line with students; if you don't understand the material, you fail. However, like you said above, the common college student entering university will not likely meet these expectations, so we must look at alternative strategies.

Here is my basic plan:

Don't get rid of the hard line approach completely. Enforce it with strict discipline with junior and senior level mathematics courses. Let this knowledge be transparent to freshman and sophomore majors, so that they are fully aware of the expectations ahead, while at the same time supplementing their education with objective tasks to transition them towards these higher expectations. This will give the student a few years to develop an academic level of maturity to be successful in the higher-level courses, or at least enough of warning to switch to a new major, like physical education, without having to stick around at college for another 2 years.

The objective tasks for freshman and sophomore students could include "gateway" examinations. For example, you said students don't have basic trigonometry, algebra, and geometry skills. Well, have your department make them learn these skills and prove it by taking a rigorous test at some point, maybe in their first semester. If they don't pass, they don't take higher level courses until they do. Of course, many students will need to review these concepts, so professors teaching intro courses should develop tools to help students accomplish this task. As another example, after the third calculus course, students are required to take a comprehensive calculus test, which will force them to go back and learn concepts they forgot.

When I returned to college, I had to take a calculus III course. One of the components of the course was a weekly lab using Maple software. Often, I found myself confused with the code as did many other students, so assignments were difficult to complete although the problems were basic in design. Graduate students ran the course and there were about 40-50 students in the lab so it was difficult to get assistance. Outside of lab, it was even more difficult, since the graduate students are extremely busy themselves. Overall, I thought the labs were a waste of time, although they had potential to be great learning tools if designed effectively. The code should already be developed so that it is easy to manipulate, which allows the student to focus more so on conceptual understanding than programming the correct code. It would also be of benefit if the labs were developed around interesting phenomena associated with the students current background of study (physics, engineering, etc.).

Hopefully, there will be more to follow when more comes to my mind and I have the time.

[buffordboy23]

Here is another thought that comes to mind. In your intro courses, assign a lot more problems as homework. Practice makes perfect, right? In a typical section of calculus books, there might be numerous ways of applying the concept. For example, there are numerous ways to compute a line integral. I see that professors assign about 10 problems per week in a math course as homework that span a few sections of the textbook, which is only a couple problems per section. So, the student doesn't get much practice solving line integrals unless they take they initiative to do it themselves. Therefore, many students probably forget how to solve these problems a short while later because they have so little experience except for the one or two examples, and likely won't even care to go back to relearn it.

So, as a professor I would assign more problems. Many of these problems would be easy munching to the student and take little time, yet they would reinforce the concepts and hopefully enter into the student's long-term memory storage. Solutions should be available to a decent number of these problems pertaining to each concept, so that students have a target to work towards and don't get easily discouraged. Hints should be given from the outright to teach supplementary ideas not necessarily discussed in the book that will appear in a certain problem and offer simplification. In this case, it may be necessary to design your own problem sets since the problems in the book may be sparse or their integrals time-consuming to calculate.

[buffordboy23]

Another idea for intro courses. We know many students don't read the textbook, so hold the students accountable. Assign sections to be read for each class. Every so often, give students a short 2-3 minute quiz about very basic ideas in the reading at the beginning of some classes. This would likely increase the number of students who are prepared for lectures and promote discipline with their studying.

[mathwonk]

those seem like good ideas. the reality we face is very serious however.

e.g. what would you suggest when faced with the following data that actually occurred to me?
My department does not give placement tests to my knowledge, so at the beginning of a calculus class, in an attempt to measure readiness, for my sake and the students', i gave a precalculus quiz, mostly covering facts on equations of lines and circles and the definition of sine and cosine.

the average score was 10%, although one Chinese transfer student got 100%. So I gave it back and made it a take home test. The next day, several students did not come back at all, and of those who did and turned in the take home test, the average score this time was 15%.

I was so discouraged I never did this again.

The only thing I know of that seems to work, is to send students to the board for presentations and detailed discussion and critique of their work, but this takes about 4-5 times as much time as is allowed in a semester.

my calculus classes think i am teaching a very theoretical class if i ask them to learn the statement of a theorem, including hypotheses, or to state a definition, although i essentially never ask them to prove anything, or even to learn a proof.

A typical incoming student has never apparently seen a definition. Even if I write
DEFINITION:...

in front of every one, almost no students, even the brilliant ones, can answer a question asking for a definition to be given. only a few think hypotheses are part of a theorem.

I am going to try to make this point this week by making up a list of series that do NOT converge even though they may appear to if you fail to check the hypotheses of the tests we have learned.

But I am worried that my students will simply give up if they are confronted with the fact that theorems are not true unless the hypotheses are true, and the need to verify this. Math has been dumbed down so much for them, some of them seem saddened by the thought that this level of precision is part of the subject.


how is it possible there is such a disconnect between what an incoming student thinks of as theory and what i think? i have even been criticized for teaching "with words".


what else should i have used to teach with?

What is going on in some high schools? teaching with pictograms?

[mathwonk]

as to gateway exams, we have turned to them in an unlikely setting: for graduate students in our phd program. we have admitted so many grad students who do not have even basic college math skills that we now have a remedial program for graduate students. and presumably it is working.

in america apparently failing students is never an option, so our educational system is built around remedial programs at every level, since students arrive in every grade without any reason to be there, except time spent in previous grades.

in your posts b23 the main thing that strikes me is that you are a returning teacher. that is a universal qualifier in my experience. i have never had a returning teacher who did not have the dedication to succeed that i look for. so maybe we could kick students out sooner, and put them in practical settings where they will learn quickly that they do need to actually know something to perform in their jobs. i.e. maybe work - study is the answer.

[mathwonk]

we tried labs for years, and they were universally considered a waste of time (as you found them) except by the people who designed them. students simply refused to put in the time needed to benefit from them. the time needed to deal with the software and hardware was overwhelming to them, and the program was a huge time sink for the few faculty who lavished enormous effort trying to make them relevant, interesting and potentially beneficial.

my feeling is that there are many things that would work on a more motivated population. so motivation is the key.

how do we motivate students to work hard, harder than they have had to in high school? to ask themselves to understand what they are doing rather than just memorize procedures?

[Tobias Funke]

What is going on in some high schools? teaching with pictograms?

I'm a high school math teacher and I ask the same thing about middle school. Like you said, these students are never held back when they should be and thus you have the problem all the way up to grad school. The problem doesn't start in high school, it starts with fractions and maybe even earlier.

I bet you won't be surprised since I've read a lot of your horror stories involving unprepared students, but about 90% of my freshmen algebra 1 class can't add fractions unless you stand over them and tell them every step. I'd say about 20% can't consistently do computations like -3-10. They simply can't remember which of 1/0 or 0/1 is undefined (and I try to explain to them why, but you must know more than anybody that they don't want explanations, just facts to remember). How can I possibly bring them from that level to the algebra they need to know in a year?

I've worked with middle school math teachers too. I was told we were covering prob and stats and I was a bit worried because stats is not my strongest area. I soon realized I had nothing to worry about because I was absolutely shocked when I saw the level these people were at. Only a few of them actually wanted to learn and the rest just suffered through the course to get more credit for a masters (in education. Many math teachers don't even seem to have a degree in math). It was embarrassing.

Of course there are the good students who make it all worth it though. A few of my calc students were genuinely excited when we took a day off of related rates to prove the formulas for the sum of the first n squares/cubes in several different ways (no boring induction), something of no relevance to the class. They actually enjoyed going through the proofs, and hopefully it will inspire them to try their own proofs.

[buffordboy23]

e.g. what would you suggest when faced with the following data that actually occurred to me?
My department does not give placement tests to my knowledge, so at the beginning of a calculus class, in an attempt to measure readiness, for my sake and the students', i gave a precalculus quiz, mostly covering facts on equations of lines and circles and the definition of sine and cosine.

the average score was 10%, although one Chinese transfer student got 100%. So I gave it back and made it a take home test. The next day, several students did not come back at all, and of those who did and turned in the take home test, the average score this time was 15%.


Wow. These scores really are dismal, even when given a second opportunity. Are these students mathematics/science majors?

EDIT: I think this would be reason enough to require placement tests. Your students are in a hole before the course even begins. In a sense, your quiz is kinda similar to a placement test. You have determined that the majority of the class is not adequately prepared for calculus, yet you are stuck with them the whole semester.


The only thing I know of that seems to work, is to send students to the board for presentations and detailed discussion and critique of their work, but this takes about 4-5 times as much time as is allowed in a semester.


This is good because it holds students accountable, but like you said it consumes a lot of time. I currently have a physics class that does something similar. Two times during the semester, each student has to give a brief 3-5 minute review of the previous lecture. The students seem to handle it well and understand the content, but, overall this consumes a total of about 2-3 whole classes throughout the semester. This further supports the idea that students are capable when they choose to be.


my calculus classes think i am teaching a very theoretical class if i ask them to learn the statement of a theorem, including hypotheses, or to state a definition, although i essentially never ask them to prove anything, or even to learn a proof.

A typical incoming student has never apparently seen a definition. Even if I write
DEFINITION:...


The thing I like about mathematics is that it is a subject that is in black and white. The basic rules are given, from which other rules follow. You always win when you play by the rules. However, I think that, in general, students don't understand the rules and why they exist. I would argue that many students find proofs very difficult, and I would wager that this is due to their high school education because they probably aren't exposed to the rigorous aspects of the mathematics.


What is going on in some high schools? teaching with pictograms?

Probably. =)

[buffordboy23]

Since my return to college, I have been very critical of the manner in which professors run their courses. I find that many of them do the basics and cover what is exactly in the book--write the theorems/definitions on the board, talk about the gist of the concept, do some examples. For a student like me, I often won't go to class like this because I read the book and learn nothing new from their lecture. I can't blame them though, since they are catering to the average student. Plus, it takes a lot of time on the part of the professor to be original with his presentation, since professors often exist in a publish-or-perish environment and have exhaustive workloads themselves.

In light of this, I thought of an alternative approach in offering a course to students. I am sure that you probably have quite a few sections of calculus each semester, with multiple professors carrying this workload. I wonder if it would be beneficial for these professors to record video-taped lectures and develop notes for each concept, so that their students can view them online. As a supplement, students can post urgent questions on a forum, where another student or another professor can respond. Essentially, it would be just like going to class, but instead, students can watch the lecture videos from their dorm and at their convenience. Of course, there needs to be student accountability somewhere in the mix for this to potentially work. There would still be regularly scheduled class to attend.

These are the immediate and foreseeable benefits:
1. Students often have trouble following the professor during a lecture, and many are scared to ask questions, so they can review the lecture over and over again as need be. This also gives more time for reflection, and the student can come to class with good and quality questions about the content.
2. Many students choose not to go to class for one reason or another, so they miss out. With this idea, students have no reason for missing a class because they can go anytime they want.
3. In the long run, this will likely reduce a professor's workload. If the video-lectures are designed effectively, they can be used semester after semester, freeing the professor from putting basic content together.
4. During the regularly scheduled classes the professor has a lot more freedom, since they don't have to cover basic content. They could discuss ways of how to think mathematically, elaborate on key ideas/theorems, introduce interesting applications, solve some examples, dedicate more class time to holding students accountable, etc.

I don't know of any college that has tried such an idea. The closest thing that comes to mind is the online courseware offered by MIT. Perhaps, my thinking is too idealistic, but it would be interesting to see how such an approach works out.

[buffordboy23]

how do we motivate students to work hard, harder than they have had to in high school? to ask themselves to understand what they are doing rather than just memorize procedures?

I think this is the fundamental problem. I think many students get discouraged easily because the work is difficult and requires time. But if they get passed through the course just by going through the motions, is intrinsic motivation necessarily an issue for them? Personally, I think many college students still need scaffolding to be successful. Should it be this way? Probably not, but this is the current state of our education system.

[mathwonk]

the video model is hard for me to imagine. i try to learn my students names and lecture with them in mind. i call upon them i class, and if i see a puzzled face i stop and ask what is bothering them. if students seem receptive i am encouraged to go beyond usual bounds and present things that are more advanced. i model the lecture on their response, and plan the next one based on questions from this class. without an audience i would be very bored and have trouble knowing what to say without the stimulation of the faces.

[will.c]

If you can expect the students to watch your lectures on their own free time, why can't you expect them to crack the damn book?

expecting students to prepare outside of class in any way is, unfortunately, not tenable - not the way basic math and science classes are expected to be taught anyway. More unfortunate yet is that these will always remain to be service classes, and so deviating from student expectations is dangerous.

[buffordboy23]

If you can expect the students to watch your lectures on their own free time, why can't you expect them to crack the damn book?


You raise a good point. I am throwing out ideas, even radical ones, to spur discussion. If one of my ideas is of any use, then I consider my time well spent.

To answer your question. Students are not held accountable to read their book. If they come to class without having read their book, then the only immediate consequence they suffer is not understanding the lecture. I believe accountability is a necessary prerequisite to improving student performance. Having the lectures already recorded will free up more time to focus on accountability. You can still find numerous ways to hold students accountable with a traditional course (e.g. presentations), but do you think more things would need to be sacrificed to do so in comparison with the video-lecture idea?

[buffordboy23]

expecting students to prepare outside of class in any way is, unfortunately, not tenable - not the way basic math and science classes are expected to be taught anyway. More unfortunate yet is that these will always remain to be service classes, and so deviating from student expectations is dangerous.

Yes, people fear change. When you take them out of their comfort level they have no choice but to adapt to a new situation. Unfortunately, to obtain some goals does require change. The widespread availability of technology in our society today has changed our society in ways unimagined in only a relatively short period of time, so why shouldn't we incorporate this in our education system? Some public schools have chosen to give their students a personal labtop computer; I don't yet know if any schools have replaced traditional textbooks with digitized copies.

EDIT: Some teachers are now required by their districts to run online chat/forum sessions with students outside normal school hours.

EDIT2: I agree that deviating from expectations can be dangerous, but only if such expectations are not conveyed clearly and maintained with consistency. This was a large reason why I saw students having difficulty in some of their classes when I was a teacher. The students did not understand their teacher's expectations on an assignment. They sometimes would ask me for assistance, and so I would look at the assignment given to them and I did not understand the expectations either.

[buffordboy23]

the video model is hard for me to imagine. i try to learn my students names and lecture with them in mind. i call upon them i class, and if i see a puzzled face i stop and ask what is bothering them. if students seem receptive i am encouraged to go beyond usual bounds and present things that are more advanced. i model the lecture on their response, and plan the next one based on questions from this class. without an audience i would be very bored and have trouble knowing what to say without the stimulation of the faces.

My original intent with the video-lecture idea was to still hold classes. If students are held accountable by completing short questions at the end of a video-lecture and submitting them, you now have foresight about misconceptions and can structure the next class meeting to rectify their misunderstandings. By holding students accountable before they arrive to class, more of them should be prepared to engage in meaningful discussion and permit you to have greater probability of success in introducing any interactive class activities. Students will still need some form of accountability sometimes during class meetings or there would be no reason to go to class. So you would still have your audience.

[CoCoA]

i gave a precalculus quiz, mostly covering facts on equations of lines and circles and the definition of sine and cosine.

the average score was 10%
This is bad enough to wonder about a systemic problem - then I saw a web log (can't find it now) from a prof in state of Wash with a child in high school algebra using a book with essentially NO algebra in it! Students "solve" problems by looking up values in tables and creating graphs.

California has a "algebra for every 8th grader" initiative (perhaps failing to legel challenges, perhpas not) that just has to water down the curriculum severely to accomplish this. Every level of our educational system just seems to be underminig the teaching of math, no wonder colleges can't get good math students.

[CoCoA]

I wonder if it would be beneficial for these professors to record video-taped lectures and develop notes for each concept, so that their students can view them online.
I took an line course a few years ago with recorded video lectures online. It was actually pretty good.

On the other hand, about a year ago I was "talked into" (forced into?) teaching a course onlilne that I thought wasn't that amenable to online delivery. It turned out to be pretty bad.

[CoCoA]

YThe widespread availability of technology in our society today has changed our society in ways unimagined in only a relatively short period of time, so why shouldn't we incorporate this in our education system?
I think schools have a wide range of effectiveness in their technology initiatives. For example, some schools have implemented well-developed online courses, others (such as one I work for, that I will not name) has several online classes that I candidly believe are nothing more than glorified independent study courses - students read on their own and simply turn in assignments to an online system. If anything, it is just another way to excuse students from attending a class. Sometimes it's justified, when some students can know the material in their sleep and just need to complete the course for the credit (it does happen in computer courses), while others need every possible interaction with the instructor to learn the material.

[buffordboy23]

California has a "algebra for every 8th grader" initiative (perhaps failing to legel challenges, perhpas not) that just has to water down the curriculum severely to accomplish this. Every level of our educational system just seems to be underminig the teaching of math, no wonder colleges can't get good math students.

I agree. One of the things I saw in public schools was the use of gimmicks to accomplish learning goals in mathematics.

Here's some examples:
1. When assisting a student with homework that required them to multiply positive single-digit integers, the student would use her index finger to tap 5 imaginary dots while performing some algorithm, called "touch-math" or something like that. I thought how strange. Why not memorize your times tables rather than some lengthy algorithm?
2. To isolate an algebraic variable, students--some of these were honor students too--would say some mnemonic catchphrase aloud or inside their head and perform some weird maneuver. I briefly tried to show them how I accomplish the same task. They were very confused and thought I was strange.
3. One of my relatives needed help with an arithmetic assignment. They had to do the problems with some gimmick method not even discussed in their textbook.

I guess the thinking that comes with these gimmick methods is that being able to compute an answer is equivalent to understanding how we could and did arrive at such answer.

[Moonbear]

The only thing I know of that seems to work, is to send students to the board for presentations and detailed discussion and critique of their work, but this takes about 4-5 times as much time as is allowed in a semester.
This is really a great teaching strategy. Getting students to DO something in a class really helps to increase learning and retention. Sitting and listening to lectures is entirely passive learning, and not very effective at all. Of course, they do need the lecture to have the material presented to them, but then taking it a step further and making the use what they have just learned is what solidifies that knowledge.

Does your course have a recitation section? If so, that is a great use for it. Instead of having a TA solving problems for students, which is really just continuing more lecture and puts them all to sleep, have the TA facilitate while the students go to the board and work the problems for one another. Then the TA will just need to watch for errors and prompt with questions when a student gets stuck (much like we do here in the HW forums here).

[mathwonk]

As an example of how much time it takes to teach, using student participation, it can take a whole period to guide a good student through a proof that the composition of two injective functions is injective.

the hard part is to get the students to connect up mentally the definitions they memorize with the steps in the proof. many students regurgitate a definition for something like injectivity, and then two seconds later have no idea how to begin a proof that some particular function is injective. (answer: begin by assuming the "given" part of the definition, i.e. take two arbitrary points in the domain, then assume either that they are different, or that their values are equal, then ......).

it is very challenging to teach learners to use quantifiers or to use them properly. letters are simply written down without defining what they mean. this is a very basic problem: the same letter is thought to mean the same thing, whereas this holds only within the same quantifier. the idea of a "variable" i.e. a letter that can have several interpretations is quite foreign.

if an injective function is defined as one such that for all x,y in dom(f), assuming x different from y, implies f(x) different from f(y), then for many students this definition cannot be readily applied in a case where the arguments have names other than x and y.

e.g. in the case of the theorem above, if f(g(a)) = f(g(b)), this does not trigger any response from the definition of injectivity of f, to conclude that g(a) = g(b). for other students this makes sense only if they rename g(a) = x, and g(b) = y, thus recovering the same names they have memorized.

somehow the teaching of algebra, i.e. the use of variables with multiple interpretations, and the corresponding understanding of quantifiers to keep book on what those variables mean, has apparently disappeared in a "saxon" high school curriculum where "algebra" means multiplying x^2 times x^3 and getting x^5.

the excellent algebra book by harold jacobs i believe, or maybe some older 60's books, treat this problem by substituting place holders like [ ], or ( ), for a variable. then the student simply fills in the box with the relevant value. this seems to help teach that anything can go in there, but two boxes of the same shape must be filled in by the same value in any given setting.

[mathwonk]

we must try to somehow maintain focus on what seems over the centuries to matter, the ability to analyze problems, to store and use prior knowledge, check ones hypotheses, and employ useful analogies.

these abstract skills seem to me what is missing, not just the rules of exponents, or the many other topics on a subject list for a specific math course.

what are some ideas for inculcating the ability to understand and use language in analyzing problems, including precise mathematical language such as variables and quantifiers?

sometimes I discuss variables as pronouns, which require antecedents just as variables require quantifiers, i.e. x is like "he", but who is he? must be specified. I take my cue here from some great old 1960's algebra books from the university of illinois i think, some of the excellent products from the 60's math revolution, like smsg books.

i still recall coming home from college as a freshman and reading these junior high books and learning the distinction between a number (abstract idea) and a numeral (concrete symbol for a number. the illustration was to imagine writing the word "milk" on the board and asking whether or not there is milk on the board. answer no, not milk but "milk" is on the board. i thought: my word - where were these books when i was in high school!?

are such books actually in use anywhere? it seems to me we do have the beginning of a solution to our problems in the existence of these wonderful materials from the 60's.

what we need next is a commitment from someone to use such materials in the schools. the political problem is how to be allowed to set a standard that not everyone may meet, or at least not without rising to it.


without the ability to require this standard, the key is motivation, how to get the child to want to learn what is actually beneficial.

my first chairman had several suggestions for motivation, something like: appeal to the beauty, or the applicability, or the historical significance, or the reliability, or the power, of mathematical results.

how much time do we spend convincing the students they will benefit from learning our subject? what are other approaches?

[buffordboy23]

I can easily tell from your posts that you are an effective professor.


what we need next is a commitment from someone to use such materials in the schools. the political problem is how to be allowed to set a standard that not everyone may meet, or at least not without rising to it.


I think there is an issue of practicality here in terms of designing a mathematics curriculum. The average student who graduates from high school will not need a rigorous background in mathematical formalism and its abstractness to be successful in the real world. However, any exposure is likely to improve their skills of logic and reasoning.

I think the larger focus of this effort should be on more academically inclined students, especially those planning to pursue the mathematics/sciences in college. But then we need effective mathematics teachers and educational settings that offer the required environment. (Before I resigned as a teacher, the administration decided to do away with honors level classes in the middle school, with one of the reasons for doing so because it distinguishes some students, the "smart ones", from others, the "dumb ones".)

It is the responsibility of the university to develop these effective teachers, and with the current level of the mathematical background of the average incoming college student, your task is a very difficult one. It may be necessary to design and add new courses to the college curriculum requirements, which focus entirely on the formalism and abstractness, since it is missing from the student's background. In conjunction, large efforts to reform our public education system are necessary to create the environment to work towards the goal of preparing these students for college. It's a cyclical process that has decayed to the current state over time.

[mathwonk]

maybe you are right that what is needed for average students is something like an old (really old) fashioned logic and rhetoric course, in which students learn and practice reasoning and argument without the obstacle of formal mathematical symbols and language.

I try to convince my students that proofs are useful in real life, e.g. in arguing for a raise with your boss, or convincing the iRS that you deserve a tax break.

e.g. the description of the conditions under which the break is given is the definition. then the theorem you set out to prove is that you qualify for it. to prove this you must address every requirement in the definition, which as you may know from experience involves lots of logical connectives like "all of the following must be satisfied, and one of the following as well..."

i may have made an error in assuming that learning to prove via rolle's theorem that a function with never zero derivative is injective, will lead to the ability to obtain a tax break or a raise.

such real life applications could help motivation. there is a psychological difficulty with offering such courses in university which i think were traditionally high school courses even in nineteenth century america (i still have my grandfather's books from the 1880's), but maybe they could be reintroduced in high schools.

[mathwonk]

im not sure how effective i am. the attention span of an average calc student is pretty short, and making calculus entertaining is tough day after day. after giving the series expression for arctan(1) = pi/4, i actually calculated it for a couple decimal places, showing how many zillion terms were needed to get good accuracy as an attempt to teach use of the error term in "taylor" series.

then i got euler's works from the library and showed how he had adapted this series using addtion formulas for tan to get over 120 places of accuracy. interestingly i also noted he made a mistake in the 112th? digit. (I actually checked them all.) that was fun for one day, but then it was what next?

and after all was said and done maybe one student could use the error term in taylor's series to show the series for e^1 converges to e, on a test.

[Nick M]

I felt that those courses in which one earns an A+ are not sufficiently challenging.

I couldn't disagree more. Most courses have learning objectives that students are expected to reach. I get straight A's because I don't miss a single lecture/lab, I complete all the readings, I do all the homework, and I take the time to think. I bust my @$$ to achieve a level of understanding that allows me to complete the work with those grades.

Why would you continuously set the bar higher and higher to the point where good students who apply themselves can't cope and begin to fail at learning? A student needs to get a B or C and leave the class confused on some subjects in order to be sufficiently challenged?

There is no excuse for students who don't show up or study, but there is also no excuse for setting someone up for failure when you are supposed to be a guiding force.

One thing I learned from attending both CC's and University is that a Ph.D. isn't synonymous for teacher. While students hold a great share of the responsibility for their own success, a "teacher" that can't connect, can't excite/inspire, and can't develop a sensitivity for the state of mind of their pupil's is really just a fogged in island of knowledge at the board. When this is layered with low expectations and tertiary factors such as language barriers, it's a wonder so many students do actually manage to progress.

And of course it starts when the person is young. I was taught statistics in high school as "Pre-Calculus". My first attempt at engineering placed me straight into Calculus I where I got a B (without any knowledge of transcendental functions), and then washed me out in Calculus II. I went to office hours, sought help, and dedicated a significant amount of time, but I had a shaky foundation that was checked off as satisfactory by those who were supposed to be my trusted advisors.

I'm now in college experience 2.0 after starting fresh in Pre-Calculus taking mathematics courses at a local CC that are at least 50% more rigorous than the ones at UMass, but are taught by people who actually know how to teach. One thing for sure is that I no longer trust any of my professors or those in leadership positions - I verify everything myself and essentially act as my own advisor and coordinator.

[Moonbear]

maybe you are right that what is needed for average students is something like an old (really old) fashioned logic and rhetoric course, in which students learn and practice reasoning and argument without the obstacle of formal mathematical symbols and language.

When I was in college, I took a course on symbolic logic, offered through the philosophy department. I had no idea what I was getting into when I registered for it, it was just one of those options to fulfill core requirements in philosophy and the only one offered that fit my schedule that term, which was usually good enough for me when choosing electives.

While I will never in my life need to remember any of the symbols used...who cares what the backwards C or upside-down U meant (if those were even the symbols used), the formal learning of how to structure an argument and to find the logical flaws, missed steps, and false conclusions was an exercise that benefits me every day. From careful designing and interpretation of experiments, to teaching material to students in a careful, logical, step-wise fashion, to arguing for or against various things people are proposing to change, etc. I think every student could use a course such as that, and perhaps something like that would be especially good preparation before taking math courses that require a lot of proofs. Basically, while they're still using words and before getting bogged down in mathematical terms, get the concept across of how, in general, a proof is supposed to function to logically demonstrate that one concept derives from another.

I know I keep injecting discussion here that's not specifically related to calculus classes, but I think it can be helpful to recognize common themes present across curricula that pertain to the modern-day students, since it will affect them in any class they take. But, it took me too long to recognize this in my own students this year, so I'm going to share it here for other's benefit, and I am going to actually take some time out next year to address it. I have noticed that a vast majority of my students still have atrocious study skills. I'm not talking about willingness to study, or laziness, or procrastination, but that when they finally sit down to study material, they really aren't studying or thinking, just reading it over and over and over hoping it will stick. These are sophomore students too. I had assumed that since I was teaching a sophomore level class and the average GPA of students admitted to this program was a 3.6, that these would be students who figured out how to study in their freshman year if they hadn't come into college already knowing that. But, somehow they skated through their first year still without acquiring those skills.

So, I'm going to take a little time out of my first few classes next year, and teach them how to study for my subject (and of course study approaches do vary a bit from one subject to the next, so even those with good study skills may need to hone them for my course). I'm already going to incorporate a team-based-learning module into the lecture, so will use those teams to first teach them the value of a study group, when a study group is done right. It takes time away from the content I can deliver, but I'd rather cover a little less content but have them learn all of it well than to cover a lot of content and have most of them only grasp a small portion of it. I'm already sitting down and outlining the learning objectives for the course for next year (nobody has done that yet for this course) and will make sure that content delivered focuses on those objectives. I'm also changing textbooks, so more of what I don't have time to deliver in lecture will be available in their textbook (it's really rough when you're stuck teaching from a book that is entirely inadequate for the course, especially when I have to point out sections that are completely irrelevant to the subject or that flat-out state things incorrectly. I also have problems that the textbook and lab guides are using different terminology for the same things...the med students can handle that there are more terms than structures because names have changed over time, but the undergrad nursing students cannot...it just confuses them at this stage of their learning). All of these are things any course can consider.

[mathwonk]

Nick, you are taking a quote that I applied to myself. I was not interested in getting A's as much as in learning as much as possible. You seem to think that failure is getting a low grade, whereas I thought of failure as not trying as hard as possible to learn at as high as level as one is capable of.

By definition an A+ means one has got all from that course that there was to get. Doesn't that make you want to see if there isn't a little more challenging course available somewhere?

I am not interested in fake awards that do not actually mean one is good. I think I told the story here once of wanting to learn to play snooker, and my method was to play against one of the best snooker players in my town every day for a year, losing every single game.

Finally I won one. After that I moved on to other even better opponents and found I had myself become one of the top players in town. Most people like "success" in the sense of winning every now and then. I didn't care about winning against patsies, to me that was not success, I wanted to beat the best, and I could stand the long apprenticeship that required.

In math getting an A+ in a non honors undergraduate class was fun for a day or two, but then I wanted to move up to the big time, and get an A in a graduate class. The truth was I didn't belong in that class I got the A+ in, except temporarily, until I got my feet under me again. In horse racing there is a concept called "dropping down in class". A horse that is used to racing in a different classification, can easily win in a lower one, even against horses with better records on paper. A professional athlete even one with no notable fame at all, will destroy amateurs at will. I wanted to elevate my classification by competing against better competition. If you go and listen to professional mathematicians talk about math, or go to lectures in a higher level course, but one in which you can understand something, you will soon be stronger than your peers who do not do this. If you read the books I recommend here, and challenge yourself as I suggest here, I believe you will soon be much stronger than you were before.

In yoga this is called the concept of fulfilling ones desires. One is motivated to go as far as his desires push him. Some people have few desires, some might say little ambition, others have much.

It seems to me you do have ambition to excel in math since you say you are taking classes now that are more challenging than the ones you took before. So I don't see you as disagreeing with me as much as you say.

[mathwonk]

wow moonbear, your dedication, insight, and positive attitude is an inspiration!

[buffordboy23]

I have noticed that a vast majority of my students still have atrocious study skills. I'm not talking about willingness to study, or laziness, or procrastination, but that when they finally sit down to study material, they really aren't studying or thinking, just reading it over and over and over hoping it will stick.

For students who try to do well, is this the only student behavior that you have observed? Or are there other common behaviors? I'm curious to know what specific study skills you will teach your students in the future, and why those methods are most important in regards to your subject. Since your an expert in your field, you have a unique perspective about how to approach your subject. I often obtained a lot of insight when the professor shared their thinking regarding their field, rather than just content knowledge.

Perhaps we could use your specific insights to give mathwonk more ideas to teach effective study skills to his students.

[Nick M]

I take failure as not understanding, which results in those poor grades. Not understanding can spawn from many things such as not having the proper prerequisite knowledge, poor teaching, or simply not making an effort. But only so much can be taught in any given semester. Just because a student succeeds in a course by getting a grade in the 95-100 range doesn't mean that they weren't challenged. I'm sure if you take this example to the extreme and say, taught the entire four semester calculus sequence that engineering/physics students take over four semesters (plus introductory linear algebra) and combined them into one semester you would wash out even the best students.

I'll be finishing up in the math department this spring with Diff-Eq (I'm majoring in EE). I've felt that we covered quite a bit in Calculus I-III and Linear Algebra. I've got all A's and A-'s, but never did I feel that I wasn't being challenged. Sure some things came easier than others, but you can only expect so much to be learned in a given time frame - especially when you have different rates of learning based on a given teaching style in a given classroom. Obviously a student shouldn't be in Calculus I/II if they don't understand how to model with transcendental functions - Nor should a student be in Multivariable Calculus if they can't comfortably work with single variables. But even with a class full of prepared and hard-working students with the same learning style and a teacher who is in perfect tune with them, only so much can be covered in a semester. What is the point of piling on more or increasing the difficulty and leaving students confused about things? I can see spending a few minutes to begin explaining something that peaks interest in material that goes beyond the scope of the course objectives, but why make the objective density so high that students have a weak understanding of 100 topics rather than a solid understanding of 50?

We used McCallum/Hughes-Hallett/Gleason for Calculus I-III and David Lay for Linear Algebra. We'll be using Blanchard/Devaney/Hall for Diff-Eq.

I like the preface from M/H-H/G regarding their vision...

"Our goal is to provide students with a clear understanding of the ideas of calculus as a solid foundation for subsequent courses in mathematics and other disciplines. When we designed this curriculum we started with a clean slate. We increased the emphasis on some topics and decreased the emphasis on others after discussions with mathematicians, engineers, physicists, chemists, biologists, and economists. We focused on key concepts, emphasizing depth of understanding rather than breadth of coverage"

The pre-calculus text from Wiley that follows the same precepts called "Functions Modeling Change" was also excellent.

Attacking subjects by developing an intuitive understanding of the underlying concepts really helps out. Then things are developed through language, and finally through exercises geared from theory and modeling. "The Rule of Four" - presenting problems verbally, numerically, graphically, and symbolically gave me a hold on things that I feel is rooted much deeper than the Calculus classes I took using another text that simply opened with proofs/theory and then hammered me with similar looking problems that just increased in mechanical complexity as the problem number increased.

I guess I have the opinion that quality is better than quantity.

[Moonbear]

For students who try to do well, is this the only student behavior that you have observed? Or are there other common behaviors? I'm curious to know what specific study skills you will teach your students in the future, and why those methods are most important in regards to your subject. Since your an expert in your field, you have a unique perspective about how to approach your subject. I often obtained a lot of insight when the professor shared their thinking regarding their field, rather than just content knowledge.

Perhaps we could use your specific insights to give mathwonk more ideas to teach effective study skills to his students.

This is not the ONLY one, but it is a pretty big problem and seems to be fairly common. Since I think someone in this thread already mentioned Piaget's learning theories, these students are still often at the concrete operational stage, where they expect a list of facts that they will memorize as facts. Their study approach focuses on that, looking at a page of notes and trying to memorize what is written there, but without really understanding it. That, and there are relationships among concepts that they are not yet making. And when the context changes...they need to use information in lecture one and relate it to lecture two in order to apply their knowledge for a clinical scenario...they can't make those connections. My clinical scenarios are probably your proofs...actually having to apply the fundamentals in a way that leads to a correct conclusion.

The other issue is they are very much still passive learners, just sitting there listening to lecture without really thinking about what is being said. When I started lecturing in the course, I tried to remedy this by having a 10 min group exercise at the end of every lecture (so, in my hour of lecture, I'd give a 50 min lecture, and then a group exercise for 10 min). This group exercise forced them to immediately use the information that had just been presented...my reasoning is that if they have to discuss it with a group, they actually have to think over an answer enough to express it to the group, and can't just sit there not thinking and waiting for the few people who did to raise their hands and provide answers at the end.

I'm considering some things like teaching them to use concept maps, which make them identify the major concepts and show how they relate to one another. The other exercise I'm considering is to have them each write a set of multiple choice quiz questions based on a particular lecture or two (due to the nature of our material, exams are usually multiple choice or short answer, not problem solving). I'm hoping this might get them into the mindset of how an exam question is constructed and what they need to think about when writing one. Multiple choice questions are harder to write for a student...it's easy for them to write a fill-in-the-blank type question without integrating concepts, but if they have to think about all the possible wrong answers they could put as distractors, then maybe it'll get them to realize how much more they need to learn than just a list of possible right answers. That sort of exercise will also help me get into their heads earlier in the term to see what they think an exam would look like.

One limitation I have is that I team teach with another lecturer. Her lectures were in the beginning of the course and unfortunately set a bad tone for the students. Her lectures would be just fine for med students, which is what she has mostly taught before, because they are already more sophisticated learners who can extract the right concepts and information from a rather dryly given lecture. It doesn't work well for the undergraduates who are still not at that level of learning. They still need repetition, big bold text that tells them something is important, and questions that prompt them to think about the connections they need to make. I will have more leeway next year, since I will take over as course coordinator. She has been attending my lectures and seeing my style and how much more responsive the students are, so is going to try to adapt some of that into her lectures too, but part of it is that her personality isn't really as well suited to that style of lecture. So, she will give her lectures as usual (maybe with some improvement), but I will ask her to keep them 50 min long instead of a full hour, then I'll come in with their group exercises at the end, since I don't think she's comfortable facilitating those yet.

[SticksandStones]

What is going on in some high schools? teaching with pictograms? Ridiculous, mind-numbingly simple regurgitation-styled questions that require no thought what-so-ever. Hour long tests that should take 10 minutes, tops (yet people still always needed extra time). Glossing over anything that might be the least bit interesting to make way for information that most of the class stands a chance at passing.

To be honest, I'm not finding college to be much different. Perhaps this is for the better though, as my study habits consist of studying maybe 5 hours a semester.

The way classes seem to go for me is this: I get into the class, get lost a bit. Study until something clicks. Class becomes easy. Rinse and repeat each semester.

EDIT: For the older people here, I have a simple question. How many hours per week during the semester did you work?

[buffordboy23]

To be honest, I'm not finding college to be much different. Perhaps this is for the better though, as my study habits consist of studying maybe 5 hours a semester.

The way classes seem to go for me is this: I get into the class, get lost a bit. Study until something clicks. Class becomes easy. Rinse and repeat each semester.


SticksandStones, I had the same experience my first time in college. The hardest part about succeeding was actually showing up to your classes and copying the professors notes. No studying was needed until the night before test day, and all you had to was memorize the notes to perform well on tests that were multiple-choice format for the most part.

My experience in returning to college (studying physics) has been very different. I have to work hard to perform well. Since I have passion for the subject, I often work harder to learn more than what is required because it will only benefit me later. A student's workload likely varies across colleges and majors.

I hear that with the number of college graduates nowadays, it is more competitive to obtain a job after college--I went on about 25 interviews and had about 5 offers for mathematics/science teaching positions. In contrast, I also hear that many employers state that newly graduated college students aren't prepared to enter the workforce. How do we rectify this trend? This underlies why students aren't prepared for Mathwonk's college calculus--there's not enough certified teachers, and of those, not enough that are effective, and academic objectives are watered-down so that intelligent students complacent with mediocrity can perform well with minimal effort.

EDIT: Consider this. During only one of my interviews for a math position, I was asked math content questions. One question that sticks out was, "What type of number is sqrt(2) and why?" Apparently, less than half of interviewees for math positions within this district were able to get the answer right.

One of the things I dislike about the structure of a typical college course is that grades are heavily weighted upon exams, and in a lot of courses, this is it. If these tests are multiple-choice, the student really learns little; he is just memorizing some facts for a test, and soon forgets them. Research projects tend to hold students to a higher level of accountability and requires them to synthesize conceptual knowledge in a unique way, which leads to a better overall understanding of the content.

[mathwonk]

heres another conundrum to muddy the water: suppose you find that your students never do anything you recommend unless you take it up and grade it, prefacing the assignment by saying "this is due thursday, and will count in your grade".

I.e. if on the other hand you say, "I will not take this up, but it is absolutely essential to understanding the topic for you to work through these problems on your own, and they will be tested on the exam", you find that most people do not do them.

What to do? You notice that those teachers who function essentially as personal trainers, making certain things "due", and worth "points", are more successful at getting their charges to hand in the work, than you are by simply telling them what is essential, and testing it on tests.


Now here is the puzzle: it seems that those teachers who get performance by "requiring" it rather than recommending it, are treating the students like high schoolers and little children. As a result, although their students may score higher on short term tests in that course, but afterwards they cannot function on their own, after leaving the environment where useful work is enforced. Is that good survival behavior in life?

I.e. is acquiring maturity a useful outcome of a course in which one is treated as if one is a responsible adult?


So what is better goal for a college course? to enforce participation in drills so as to get a higher score on a narrowly based test? or treat students as adults, to help them learn the consequences of self motivated and self disciplined learning?

If acquiring maturity is desirable, how does one encourage this, so as few as possible fail out before taking charge of their own learning? I am interested in how to produce more people with a desire and willingness to do what they know will help them, without it being enforced by artificial means like meaningless "points".


This is partly because I do not believe it is possible to make anyone think deeply, or understand something they refuse to try to understand.

So how do we get our charges to begin asking themselves, not "have I got all the points available here?", but "have I really understood this? could I answer other different questions using this same idea?"

When I ask a student why he/she thinks a certain argument is a proof, I often get the answer "that's what you said" or "that's what it said in the book". I try to make the point that a valid proof is one that makes sense, not one that mimics almost faithfully (merely omitting the definitions, the logic, and other key ingredients), the words found in another source.

How do we teach that the validity of an argument is measured not by comparing the words with those in a book, but by considering the meanings of those words?


the last paragraph of buffordboy23's previous post is key. Is it possible that for many of us this behavior is only acquired by leaving school? Thats what did it for me.

[mathwonk]

anyone curious as to the value of gre scores in deciding admissions will get some insight from the previous remarks. We want students who will try to get to the bottom of things, not merely ones who can compute accurately the area between two curves.

[Moonbear]

heres another conundrum to muddy the water: suppose you find that your students never do anything you recommend unless you take it up and grade it, prefacing the assignment by saying "this is due thursday, and will count in your grade".

I.e. if on the other hand you say, "I will not take this up, but it is absolutely essential to understanding the topic for you to work through these problems on your own, and they will be tested on the exam", you find that most people do not do them.

What to do? You notice that those teachers who function essentially as personal trainers, making certain things "due", and worth "points", are more successful at getting their charges to hand in the work, than you are by simply telling them what is essential, and testing it on tests.


Now here is the puzzle: it seems that those teachers who get performance by "requiring" it rather than recommending it, are treating the students like high schoolers and little children. As a result, although their students may score higher on short term tests in that course, but afterwards they cannot function on their own, after leaving the environment where useful work is enforced. Is that good survival behavior in life?
Yes, it feels like treating them like little children, but some really are still at that level of maturity. Some of it is also that they are trying to prioritize the workload of all their courses. Most of my students will eventually get to those exercises they are told will help but that are not collected, pretty much the day or two before the exam. Of course, if they worked on it when it was recommended, everything else afterward would have been easier for them to learn. This is all part of the problem of study skills, and perhaps time management...their inefficiency makes it more of a struggle than if they just studied a little bit at a time efficiently.

On the other hand, for the vast majority of jobs, this is perfectly fine survival behavior. If you're the boss or a project manager, yes, you need to have the work ethic to look into every aspect of a project if you're going to do well, but if you're just the employee, then all you need to be able to do is what your boss tells you to get done by the deadline they set. When you see a student going the extra mile on their own, they are the ones to encourage to consider grad school. For those heading out into industry, doing what needs to be done "to count" and by a deadline is really all they'll ever need to do.

So what is better goal for a college course? to enforce participation in drills so as to get a higher score on a narrowly based test? or treat students as adults, to help them learn the consequences of self motivated and self disciplined learning?
I prefer to enforce participation on things that will teach them the bigger picture. If they can grasp a few major concepts, even if it's through persuasion, trickery and bribery, then they will walk out of my classroom with the minimum skill set needed to go back and learn the details on their own later when they need them.

If acquiring maturity is desirable, how does one encourage this, so as few as possible fail out before taking charge of their own learning? I am interested in how to produce more people with a desire and willingness to do what they know will help them, without it being enforced by artificial means like meaningless "points".
If you can show your students how a particular topic is relevant to their goals beyond just passing your course, then it's easier to "hook" them into learning out of self-interest. This is why in a nursing course, I use clinical cases to reinforce major concepts. Whenever possible, I try to adapt real cases involving situations that they might encounter outside the narrow confines of whichever doctor's practice they end up working in. For example, I used one of someone stumbling and vomiting at a conference at a hotel to illustrate how their knowledge of nervous system anatomy would help them realize this person was NOT just drunk, but having a stroke, and that they could even figure out where that stroke was happening by his behaviors. So, even if they end up working in a podiatrist's office, they see the relevance of understanding what's happening in the brain to know better than the average person when to call an ambulance rather than telling someone to "sleep it off."

If you can relate the subject to their interests in life, you'll have them hooked. Heck, sometimes I just point out things like, "...and you should be sure to understand this well and remember it, because most of your physiology course next semester will be based on this part of the nervous system." Now they know they need to do more than remember it long enough to take my exam, but also need to remember it to do well in another entire course coming soon.

This is partly because I do not believe it is possible to make anyone think deeply, or understand something they refuse to try to understand.
You can make someone think deeply. You can't make someone understand what they refuse to understand. I don't worry a lot about those who make no effort whatsoever. If they have no interest at all in learning the subject, the best thing that could happen to them is they fail the course and wake up to the fact that they have chosen the wrong major. But, for those who are simply not yet sophisticated learners, we can challenge their thought process.

So how do we get our charges to begin asking themselves, not "have I got all the points available here?", but "have I really understood this? could I answer other different questions using this same idea?"
This is what I use the group exercises at the end of class to do. In fact, in my very last lecture, I gave them a case that had no answer (or at least not a complete answer until additional diagnostic tests are added that would go beyond my course content). This is much more the reality they will see in the working world. They may not have all the answers, but they need to know enough to recognize there's a problem and the general system being affected so they know when to call the doctor in, and what to do while waiting for the doctor to return their call, etc. I told them if they could answer all the questions I asked related to that case, they were ready for the final exam, because it made them use everything they had covered not only in the lectures included on this exam, but even topics that were covered in their very first lecture of the course. A few students grasped it well and showed they really understood what was going on. More of them realized how little they still understood about putting the concepts together, and that case prompted a lot of emails and office appointments. A few walked out with no interest in learning about the case...they are the ones who have been consistently doing poorly in the course and I'll probably see again next year.

Sometimes, I catch a few more students during exam review sessions. My format this time around was to first let them ask any questions they had. Then when they had no more to ask, I started asking them questions. Again, some students had grasped the concepts and were able to answer my questions readily, while others could do it with some thinking, and then there were some still sitting there just trying to write it all down because they were realizing they still had a lot to learn. I then let them ask questions again, in case my questioning prompted them to think of something else that had them confused. I've gotten yet another round of emails and requests for office appointments after the review session. It may have taken me until the very last day of class to finally get through to them what they need to learn and how they need to approach learning, but while it would have been nice for them to come in able to do this, I feel satisfied that I've done my job well if they at least walk out of the class with that knowledge in hand.

When I ask a student why he/she thinks a certain argument is a proof, I often get the answer "that's what you said" or "that's what it said in the book".
I get that too. In my review session, when I asked a student to give an answer and then asked them why they chose that answer, they said, "Because that's what's in your notes." So, I simply just don't let them off the hook at that point. I ask them why they should believe my notes, or why I would write that in my notes.

They've heard me repeat a phrase in many, many, many of my lectures. "You shouldn't believe me." This is the key to critical thinking. They should not take what I tell them on faith, they should prove it to themselves. If they can prove it to themselves, THEN they understand it. This is sophisticated learning, and fairly advanced expectations. I don't expect all of my undergraduate students will attain this level, but I want them to start having that seed planted so that by the time they're reaching their senior year, or considering graduate or professional schools, that seed has finally grown into understanding. Someone needs to plant the seed, but then I have to put faith in the other faculty they have after me that they will continue to water and nurture that seed or seedling until those students fully blossom into mature thinkers. This isn't a process that happens overnight.

[SticksandStones]

Now here is the puzzle: it seems that those teachers who get performance by "requiring" it rather than recommending it, are treating the students like high schoolers and little children. As a result, although their students may score higher on short term tests in that course, but afterwards they cannot function on their own, after leaving the environment where useful work is enforced. Is that good survival behavior in life?
Well, the way I see it from a student's perspective is this. High schools treat their students like elementary school children. They're more concerned with whether or not you're walking in the halls and using "Mister" and "Misses" when addressing a teacher than whether they are capable of finding the roots of a quadratic equation.

From what I've heard from people who attended high school in the 70's (and earlier), and from people who've attended high school in foreign countries (even recently), the high school education here is pathetic in comparison. If you want to know why you're students are not prepared for your class and why they have no work ethic it's because they've spent the past 13 years of their life being babied.

[Joskoplas]

If you want to know why you're students are not prepared for your class and why they have no work ethic it's because they've spent the past 13 years of their life being babied.

And the teachers chose to do it this way, eh? No pressure from anyone? No mandates to serve all and please everyone? No forced accommodation for everyone's long list of disabilities, long litany of excuses, or long line of helicopter parents?

It's easy to throw stones at the perpetual scapegoats, but I for one was inspired to a career in science by an excellent, no-compromises chemistry teacher.

[SticksandStones]

And the teachers chose to do it this way, eh? No pressure from anyone? No mandates to serve all and please everyone? No forced accommodation for everyone's long list of disabilities, long litany of excuses, or long line of helicopter parents?

It's easy to throw stones at the perpetual scapegoats, but I for one was inspired to a career in science by an excellent, no-compromises chemistry teacher.

Where did I say it was the teacher's exclusive fault?

I made a point, that you seem to agree with. That the High Schools are failing their students. Whether this is the fault of teachers, parents, administration, or all of the above is another matter.

[Joskoplas]

Don't read it as an accusation; I was inputting a second perspective. You may feel, as a student, that your world sucks. There's a big list of reasons why things came to be the way they are, and a Gordian knot to untie to undo the damage.

[SticksandStones]

Don't read it as an accusation; I was inputting a second perspective. You may feel, as a student, that your world sucks. There's a big list of reasons why things came to be the way they are, and a Gordian knot to untie to undo the damage.

What the hell are you going on about?

Nowhere did I say my world sucks. My world is great, thank you very much.

I offered one explanation for why mathwonk, et al, are finding their students being incapable or unwilling to do the work necessary to succeed and you come in with what appears to be an attempt to counter my explanation by countering points I didn't make with points that only go to further strengthen my explanation.

So, again, what are you talking about?

[Joskoplas]

Well, the way I see it from a student's perspective is this. High schools treat their students like elementary school children. They're more concerned with whether or not you're walking in the halls and using "Mister" and "Misses" when addressing a teacher than whether they are capable of finding the roots of a quadratic equation.

Like you, I read more into it than was necessary. I'm out now. So I guess I'm not 'the hell' going on about anything.

[CoCoA]

Well, the way I see it from a student's perspective is this. High schools treat their students like elementary school children. They're more concerned with whether or not you're walking in the halls and using "Mister" and "Misses" when addressing a teacher than whether they are capable of finding the roots of a quadratic equation.

I would never be a high school teacher, but if I were and a student called me by my first name instead of using Mr., I would probably throw the student out of class. (Not because he/she is a student, but because he/she is a non-adult addressing an adult - and an authority figure at that - in a disrespectful manner. I was taught better by my parents)

I have taught a lot of community college classes, and I never insist my students call me mr. - after all, they are adults by then - but invariably they do. I take classes at university, and I call my professors Dr., even though I am older than some of them.

From what I've heard from people who attended high school in the 70's (and earlier), and from people who've attended high school in foreign countries (even recently), the high school education here is pathetic in comparison. If you want to know why you're students are not prepared for your class and why they have no work ethic it's because they've spent the past 13 years of their life being babied.
When I went to school, I walked 15 miles through the snow, uphill both ways. Oops - gotta go, I have to tell some kids to get off of my lawn.

[SticksandStones]

I would never be a high school teacher, but if I were and a student called me by my first name instead of using Mr., I would probably throw the student out of class. (Not because he/she is a student, but because he/she is a non-adult addressing an adult - and an authority figure at that - in a disrespectful manner. I was taught better by my parents) I don't see how what year one was born in has any baring on what one calls someone, but that's just me and to each his own.

In my high school's case, given the number of students in my graduating class alone that couldn't read at a satisfactory level I think the disrespect towards the [horribly inept] teachers was warranted.

I mean, let's put it this way. They weren't doing their job. They were and are wasting tax payer money, wasting my time, and hurting a generation of students by being incompetent, lazy, and in some cases just down right stupid. Why on Earth would I call someone "Mrs. So-and-So" when their idea of teaching history is to hand out a list of words we need to memorize by the end of the month and then spend the 4 weeks sitting on a computer looking at clothes?

Now, let me clarify something. I don't think high school students should just off-the-bat ignore and disrespect their teachers. However, I think that it's the teacher's responsibility to do their job and show that they are deserving of respect.

Which, again, if you want to know why students in college are becoming more and more lazy the above is a reason.
I have taught a lot of community college classes, and I never insist my students call me mr. - after all, they are adults by then - but invariably they do. I take classes at university, and I call my professors Dr., even though I am older than some of them.
Most of my professors have insisted on us calling them by their first name, but on the first day I'll usually refer to them as "Professor" or "Doctor so and so".

When I went to school, I walked 15 miles through the snow, uphill both ways. Oops - gotta go, I have to tell some kids to get off of my lawn.
Never did that in high school, but that's roughly my college experience. :)

[Tobias Funke]

I know I sort of complained about the students mostly in my last post, and I hope I don't come off as too negative (I'd offer helpful suggestions but I'm just a 2nd year teacher trying to learn from you guys), but I wasn't kidding when I said that most of the middle school teachers I worked with were embarrassing.

Most didn't even remember the triangle or unit circle definitions of the trig functions, many couldn't make the leap from 1+2=2(3)/2, 1+2+3=3(4)/2, 1+2+3+4=4(5)/2, and maybe even one more line, to guess 1+2+...+n , and some couldn't even understand the very concept of generalizing to arbitrary n. One (and likely many more) couldn't do basic exponent problems like (x^2z^(-1))/(2z^3) or solve a simple "real world" geometry problem, something about finding the cost of carpet to cover a room given the cost per square foot, from her students' book. Sure, teachers may be ridiculously underpaid and have to put up with a broken system, but that has nothing to do with the students.

So here we are, myself included, talking about our students' deficiencies, but are they really that surprising? I know exactly where they come from.

[Moonbear]

I know I sort of complained about the students mostly in my last post, and I hope I don't come off as too negative (I'd offer helpful suggestions but I'm just a 2nd year teacher trying to learn from you guys), but I wasn't kidding when I said that most of the middle school teachers I worked with were embarrassing.

Most didn't even remember the triangle or unit circle definitions of the trig functions, many couldn't make the leap from 1+2=2(3)/2, 1+2+3=3(4)/2, 1+2+3+4=4(5)/2, and maybe even one more line, to guess 1+2+...+n , and some couldn't even understand the very concept of generalizing to arbitrary n. One (and likely many more) couldn't do basic exponent problems like (x^2z^(-1))/(2z^3) or solve a simple "real world" geometry problem, something about finding the cost of carpet to cover a room given the cost per square foot, from her students' book. Sure, teachers may be ridiculously underpaid and have to put up with a broken system, but that has nothing to do with the students.
That is really quite sad, what you've described, and what SticksandStones has described in his elaboration of his problems with his teachers.

Honestly, I had no idea it was so bad! At the university level, we don't meet many middle and high school teachers, we just see the end product in our students. The few teachers we do meet are most certainly among the cream of the crop, because they're the ones arranging for their classes to visit us and to learn something above and beyond the usual curriculum. For example, we had a teacher from the local high school who teaches a first aid course bring her class in to learn some anatomy from the anatomy faculty here. It was great fun for both us and the students, and that's certainly not a teacher slacking on her work.

So here we are, myself included, talking about our students' deficiencies, but are they really that surprising? I know exactly where they come from.
How do you think it got so bad? Is it a lack of qualified teachers that students have given up on their teachers, or is it that the students had become so disrespectful that the teachers gave up on them first?

And, more importantly, how do we fix it? In part, the things we're discussing in this thread are a start, not to solve the problem, but to address it once it falls into our laps at the university level. And, one of the visions we had for this particular subforum here was to also help with some of these issues, in terms of those of us with university teaching expertise to reach out and communicate with those with middle and high school teaching expertise to figure out how to bridge the widening gap students are needing to leap as they enter college.

Though, I wonder, what else can we do? Is there more of a need for outreach activities, of university faculty visiting high schools and giving a brief lecture about what we do and why it's worth learning?

What happens in teacher's workshops and what sort of continuing education do they get? Is there a place for university faculty beyond the education departments to offer workshops to teachers to refresh and update their knowledge in our subjects? When I was in grad school, one of the colleges at our university had an annual "Teacher's Day" in which teachers from around the state were invited to participate in seminars and workshops led by university faculty. Because I was doing some unique work with running study groups to teach students to use study groups, one of the faculty invited me to join him to give one of the presentations on using small group learning in science courses. He was taking a different approach to incorporating it into our lab courses. Afterall, our Freshmen weren't all that different from the high school seniors, so what we were finding worked with them was certainly applicable to at least a senior level high school class.

[CoCoA]

Why on Earth would I call someone "Mrs. So-and-So"

Don't ask me that one, ask your parents.

[tshafer]

How do you think it got so bad? Is it a lack of qualified teachers that students have given up on their teachers, or is it that the students had become so disrespectful that the teachers gave up on them first?
What's crazy is this isn't just freshmen. I've taken graduate mathematics classes as an undergraduate here, and you wouldn't believe how whiny some of these graduate students are. They have become accustomed to being handed some simple work which they can finish quickly and then get a decent grade with no extra effort. Attend a colloquium? You must be joking. No extra credit assignments? The horror. Read the book ahead of and after class to understand the lecture? Not happening. It's all really quite unbelievable.

Of course, I am not too dissimilar. Being a senior hoping to get into graduate school, it was only last year I would wager that I learned to buckle down and focus. I've gotten straight As all throughout my physics and maths courses with a minimum of effort -- most courses are rehashes of the texts -- and I assumed that I was learning everything I could. Up until last year, when I was finally convinced that going to graduate school was a viable option (frankly, it had never occurred to me!), it was easy to go through my E&M courses, Mechanics courses, etc., and do enough to get As without a deep knowledge of material. Now I am seeing the error of that way of operating and it has markedly improved my grasp of material this year. I just wish someone had smacked me upside the head as a freshman and said that grades weren't enough so that it didn't take until end of junior year to grasp it.

And then this year I had my first truly difficult class (graduate ODE with the one truly excellent professor I've taken) and I aced it -- but I had to earn it this time. Lots of work, lots of study time, but holy cow -- I know ODE better than I know most things I've been taught. One of the better parts of my undergraduate career.

[mathwonk]

i sent my kids to the best school quite possibly in the state. i borrowed money to pay tuition. the first day i was worried my son was not dressed up sufficiently. then i saw his teacher was wearing sweat pants. the kids all called he teachers by their first names. some years later this became the first school in georgia to win a national math contest.

being hung up on what name kids call you is a sure sign of not having even a clue about what matters in teaching, unless you are teaching military style, i.e. mindless, obedience instead of thinking. if you earn respect you will get it voluntarily. if you demand it, you will be made fun of behind your back. if your students love you they will want to work at what you recommend, if you enforce discipline, they will try to avoid it.

[buffordboy23]

Moonbear,

Thank you for the great feedback. Students should feel fortunate to have professors like Mathwonk and you. It's easy for anyone to be perceptive of the problems in education, but taking initiative in trying to find and apply solutions is another matter.

This is not the ONLY one, but it is a pretty big problem and seems to be fairly common. Since I think someone in this thread already mentioned Piaget's learning theories, these students are still often at the concrete operational stage, where they expect a list of facts that they will memorize as facts. Their study approach focuses on that, looking at a page of notes and trying to memorize what is written there, but without really understanding it.

Unfortunately, this basic level of understanding seems to be all that is required in a lot of educational settings. One of the previous posters talked about how they had to memorize vocabulary words for some class, which didn't seem to be on par with what typical learning objectives should be. This is the fault of the teachers no doubt. I often think that we (educators) focus more on obtaining the right answer rather the originality and creativity associated with an approach to solving a problem. The interesting problems are open-ended, but yet most of the focus is on the absolute.

The other issue is they are very much still passive learners, just sitting there listening to lecture without really thinking about what is being said. When I started lecturing in the course, I tried to remedy this by having a 10 min group exercise at the end of every lecture (so, in my hour of lecture, I'd give a 50 min lecture, and then a group exercise for 10 min). This group exercise forced them to immediately use the information that had just been presented...my reasoning is that if they have to discuss it with a group, they actually have to think over an answer enough to express it to the group, and can't just sit there not thinking and waiting for the few people who did to raise their hands and provide answers at the end.

Research shows that when students participate actively in a lesson, their overall retention of the concepts improves drastically. Closure activities, like what you incorporate, are extremely important as well and improve understanding. These results are in regards to K-12, but I am sure that they hold much weight as well in a college setting.

[buffordboy23]

That is really quite sad, what you've described, and what SticksandStones has described in his elaboration of his problems with his teachers.

Honestly, I had no idea it was so bad!

My teacher co-workers were very competent in regards to their content knowledge, so I don't know if Tobias Funke's comment is a local or global problem. His comment specifically targets the competency of math teachers. It is known that there is a shortage of certified math teachers. Certification does not necessarily mean competency to teach the subject matter either, so mathematics is a large problem area in our education system. The certification tests are produced by the same company that produces the GRE exams.


What happens in teacher's workshops and what sort of continuing education do they get? Is there a place for university faculty beyond the education departments to offer workshops to teachers to refresh and update their knowledge in our subjects?

I worked in the Pennsylvania education system. The last I knew, teachers are required to obtain 24 credit hours of continuing education within five years after graduation, or else their teaching certificate is removed until the requirements are met. Moreover, every five-years teachers have to complete 150 hours of continuing education, which can include workshops or college courses, or face the same penalty.

University/K-12 collaborations would probably be beneficial, and this happens in many locales. I have never participated in one so can't comment on specifics.

Another problem I see with the K-12 education with the mathematics/sciences is a lack of good resources. This is true particularly for textbooks. AAAS Project 2061 has given many of the middle school science textbooks in usage a failing grade; I think the same holds true for mathematics textbooks but can't say unequivocally. Unfortunately, this seems to be the backbone of a teacher's curriculum, since many schools (mine, for example) could not afford science kits backed by educational research. With all of the requirements placed on a teacher, especially new teachers, it can be difficult for them to find time to remove this crutch of relying on textbook.

EDIT: The workshops that I have participated in have gone both bad and good. A lot of it is dependent upon the presenter. I participated in numerous Learning-Focused Schools workshops:
http://www.learningfocused.com/
This education model has been successful in a number of schools. The metric for the research studies on the model has been the federally-mandated tests. The validity of these standardized tests seems to be questionable, so its hard to say how effective the model it really is. Moroever, the research studies were done by who else, but the organization who is selling the products. There used to be a link that showed the research results (it may still exist, but it's been awhile since I used their website and they updated their web-layout) and my initial thoughts were biased statistical reporting. Nevertheless, it the methodologies I acquired seemed to have an impact on my success in the classroom.

What did we do in the workshop? Well, first we had to take time off from our teaching duties, requiring a substitute for 3-4 days over the school year. We learned an overview of the educational model, with certain components delegated to certain days of the workshop. We would learn the methodologies from the presenter, who modeled with examples. This was followed by collaboration with fellow teachers and we practiced applying these ideas to our particular context in the classroom. It was recommended that the school administrators, some of who were present at the workshops, form committees with teachers and monitor progress with the methods learned until next workshop. Overall, it was the best workshop that I participated in. There was easy and direct application of these ideas into our lessons and accountability on the teacher. Such is not always the case with workshops.

[Moonbear]

I worked in the Pennsylvania education system. The last I knew, teachers are required to obtain 24 credit hours of continuing education within five years after graduation, or else their teaching certificate is removed until the requirements are met. Moreover, every five-years teachers have to complete 150 hours of continuing education, which can include workshops or college courses, or face the same penalty.
Is that universal, or does it vary state-by-state?

Another problem I see with the K-12 education with the mathematics/sciences is a lack of good resources. This is true particularly for textbooks. AAAS Project 2061 has given many of the middle school science textbooks in usage a failing grade; I think the same holds true for mathematics textbooks but can't say unequivocally. Unfortunately, this seems to be the backbone of a teacher's curriculum, since many schools (mine, for example) could not afford science kits backed by educational research. With all of the requirements placed on a teacher, especially new teachers, it can be difficult for them to find time to remove this crutch of relying on textbook.[quote]
That's consistent with what I've seen posted around here from parents concerned about the books assigned to their students, and looking for supplemental materials. The topics being covered in the books seemed incredibly inadequate.

[quote]What did we do in the workshop? Well, first we had to take time off from our teaching duties, requiring a substitute for 3-4 days over the school year.
Do school districts still typically schedule teacher's workshop days into the school calendar? I'm wondering if this is in addition to those days, or if schools are no longer supporting this requirement. What I mean is that when I was a kid, a couple times a year, we'd have a day or two off from school as a teacher's workshop day. That meant teachers could participate in these required workshops without getting substitutes. If schools are expecting all time for teacher's workshops be done during hours when school is in session, and that they need to take time away from their classes and find substitutes to complete those requirements, then this seems detrimental to the students and a poor message to send to the teachers about how much the district really supports their continuing education.

[mathwonk]

one of the best ways to improve knowledge of subject matter i know of is to attend the summer program at park city, the pcmi park city math institute run by the institute for advanced study. There, high school teachers, undegrads, grads, and prefessional mathematicians, i.e. college professors, all live and meet in the same locale, have lunch together, and attend workshops of their choice. people are encouraged to attend workshops oriented to others spoecialties and to converse about shared problems.

At these meetings I have sat in on sessions for teachers as well as undergrds and grad students, plus those for researchers in my area. There are also evening presentations intended to address a question of interest to everyone in a way all can enjoy. Then after the summer session, groups from the same geographic area go home to continue in some way the activity with followup during the winter.

I have also given teachers my notes on graduate algebra and galois theory, hoping some will find a way to work this stuff into honors level classes for bright students. I have also given materials to grad students studying for prelims at other schools and had some feedback that they were helpful. This is a great place to learn and to meet interested learners of math, probably the best i know of. the program this summer is on arithmetic of L functions, i.e. number theory at the highest level. the legendary john tate will be there, my calculus teacher from 1960, and still active in research.

The quality and level of the presentations is so high I often find the graduate classes are about right for me. They also issue books afterwards recording some of the presentations. Check out IAS - PCMI.

or this link:

http://pcmi.ias.edu/current/program.php

[mathwonk]

here is a blurb from the education section of the summer program for 2009:
The SSTP is structured around three goals:
All teachers should be involved in
• continuing to learn and do mathematics
• analyzing and refining classroom practice
• becoming resources to colleagues and the profession.

Each of these goals is reflected in the three strands that comprise the summer courses and activities.

Some Questions and Problems in Arithmetic (2 hours per day, 5 days per week)
This course will investigate questions like these:

In how many ways can an integer be written as the sum of two squares?
In how many ways can an integer be written as the sum of four squares?
What's the probability that an integer picked at random has no perfect square factor?
What's the probability that two integers picked at random have no common factor?
Which linear functions f(x) = ax+b (a and b integers) generate infinitely many prime numbers for integer values of x?
What is the probability that an integer picked at random between 1 and 1020 is a prime number?
The real goal of the course is to answer the following quesiton:

How are all of the above questions related?

[buffordboy23]

Is that universal, or does it vary state-by-state?

I don't recall. I'll have to research it. Tobias Funke, are your state standards similar to those in Pennsylvania?


Do school districts still typically schedule teacher's workshop days into the school calendar? I'm wondering if this is in addition to those days, or if schools are no longer supporting this requirement. What I mean is that when I was a kid, a couple times a year, we'd have a day or two off from school as a teacher's workshop day. That meant teachers could participate in these required workshops without getting substitutes. If schools are expecting all time for teacher's workshops be done during hours when school is in session, and that they need to take time away from their classes and find substitutes to complete those requirements, then this seems detrimental to the students and a poor message to send to the teachers about how much the district really supports their continuing education.

My district scheduled in-service days into the calender, which may not be synonymous with what you had in mind. At these events, we would learn about new district policies, other odds-and-ends, and occasionally, a presenter would lead a mini-workshop. I think it is common for teachers to get off from school to go to workshops, and I agree that it affects the students' education. However, the federally mandated tests (No Child Left Behind) likely affects students' education more--there is a two-week testing window (so throw all your lesson plans for those two weeks of testing out the window), plus the the objectives for a course often become objectives seen on the "big" test.

[yeongil]

I teach math and music (I know, interesting combination) at a all-girls catholic high school. I am certified in math only (not in music -- in fact, many of our teachers are not certified). I am having the same issues as Tobias Funke, in that our entering freshmen have problems with fractions and negative numbers. They are also way too dependent on their TI-83/84's for computation. I teach the honors section of Algebra I, and I actually forbade those students to use the calculator for tests and quizzes in most of the first quarter. In some cases the results were not pretty.

Our math dept. has been complaining for years that we're admitting too many students who are not ready for high-school level math, but obviously the reason is that the students who do well in math in middle school are going to public school or being taken by other higher-performing private schools. Furthermore, it looks like that the math skills possessed by the elementary/middle school teachers in our feeder schools are lacking.

I can't tell you the number of times students write the wrong answer because they inputed into the calculator wrong. Two of my favorites:
(1) students will say something like 80/40 = 1/2 because they divided 40/80 in their calculators (they switch the order of numbers). Too many think that a number divided by 0 is 0 because they'll type 0 divide by (number) in their calculators.
(2) students will say something like (4 + 8)/2 = 8 because they forget to type the parentheses -- 4 + 8/2 = 8. They're not mindful of the order of operations.

I wish that calculators were forbidden to high school students until maybe Algebra II and above, where the graphing features may be useful. But I think I'm in the minority amongst high-school math teachers.

I also teach honors Precalculus. Some more favorite errors I've seen:
(1) (x + 4)^2 = x^2 + 16. Students forget to foil.
(2) ln x - ln y = (ln x)/(ln y). Students forget to use only one ln for the division, as in ln(x/y).
(3) I still get students who think that (x^3)^3 is x^6 and not x^9!!!!!!

I can go on and on, but I won't, because this post is already depressing.

Is that universal, or does it vary state-by-state?
I was certified 3 years ago for the state of Maryland. In Maryland I know that I have to take 6 credit hours within 5 years to maintain certification. I don't know if it's 6 credit hours every 5 years, of if I'll have to take more credits within the second 5-year period.


01

[mathwonk]

your stories do not either surprise or depress me. that is just a small verification of the 30+ years of examples i have under my teaching belt. i agree with everything you say by the way, calculators should be smashed and burned in a pyre, until the student learns how to use them.

[mathwonk]

I am up at just after 6 am to go in today and offer a free day (all day) of review to my classes. Seeing this title reminds me of the changes since the old days. When I was student there was in 4 years of college never a single hour of review offered by any of my professors. Moreover the last week of class this semester has been spent repeating things I have already taught 2 and 3 times before. Nothing in my classes as a student was ever repeated. Every lecture was new material, and that continued up through the last day. In the classes I teach many students persist in trying, and unfortunately now succeeding, to get me to do for them what they should do for themselves, i.e. look up and read basic material in the book, review class notes, make up practice exams. I try to teach my classes things no one ever taught me: how to study, how to review, how to make up a sample test and exam. I am not sure I am having any success. But we continue to try. It does work for a few, and maybe more slowly for some others. I recall I also was an unresponsive student for a long time, and now as an old man I appreciate things that teachers did, that they never realized I appreciated at the time. It just takes some of us longer to become responsible for our own learning. The key is not becoming discouraged when you see little progress from your efforts.

[Nick M]

My math teachers plan one snow day into our schedule. If at the end of the semester we have had no snow days, then we use the last day for review.

[Cincinnatus]

Originally Posted by buffordboy23 View Post

I worked in the Pennsylvania education system. The last I knew, teachers are required to obtain 24 credit hours of continuing education within five years after graduation, or else their teaching certificate is removed until the requirements are met. Moreover, every five-years teachers have to complete 150 hours of continuing education, which can include workshops or college courses, or face the same penalty.

Is that universal, or does it vary state-by-state?

Continuing education for K-12 teachers is in trouble in New York state. Due to the current financial crisis they are probably going to be making extreme cuts to these programs. In particular, the New York State "Teacher Centers" which are quasi-local state-funded organizations have been targeted to receive either major budget cuts or to be eliminated entirely. These Teacher Centers are responsible for coordinating the majority of professional development options for K-12 teachers in New York state.

If anyone reading this is from New York, you should write to your state senator or the governor's office on behalf of professional development and continuing education for teachers. They are supposedly working on the budget now and will announce it next week.

[Tobias Funke]

I don't recall. I'll have to research it. Tobias Funke, are your state standards similar to those in Pennsylvania?

I'm in MA, so probably. I don't know for sure because I'm in a private school. But anyway, "continuing education" isn't synonymous with learning actual mathematics. These teachers I spoke of were in classes like that, and they were basically guaranteed to pass. They learned a little, but it was material they should have known. But mathwonk's courses seem better than these, so I may be wrong.

[mathwonk]

University of Georgia is apparently very unusual. Our math ed department requires its students, in addition to their work in methods and theory of instruction, to take math classes in the math dept. In particular they take some of the same courses that are required for math majors, such as courses in proofs and abstract algebra.

This is part of the reason the UGA math ed. department stands out nationwide, according to a recent survey which found it the only "exemplary" math ed. dept out of 77 programs studied in the country.

"UGA stands out because the program has stringent requirements and stresses the importance of conceptual knowledge, according to Denise Mewborn, professor and chair of the College of Education’s department of mathematics and science education.

“The big emphasis for the past 10 to 15 years has been on developing conceptual understanding in children, not just teaching them procedures, rule without reason,” said Mewborn. “Getting them to understand why these things work so they’re not just playing Russian roulette.”

Prospective elementary education students at UGA are required to take three content and two methods courses, while many schools only require one method and two content courses.

Mewborn also pointed to the close relationship between the elementary math education program and the UGA mathematics department in the Franklin College of Arts and Sciences as another reason why the program excels in preparing teachers for the classroom."

http://www.uga.edu/columns/080902/news-math.html

[mathwonk]

Offering good continuing education is an ongoing struggle with many challenges, some financial some political. A number of years back I taught such a course for returning teachers in calculus. I believe I used Apostol, the famous book for honors calculus students at schools like MIT and Stanford. At the end of the course I gave copies of the book "How to solve it" by Polya, to the members of the class.

It was one of my best experiences all round in a classroom. One of the teacher/students gave me a poem about her view of teaching. This person later was named the top teacher in Georgia one year.

But the class was small, and a few students dropped out, some complaining bitterly. The enrollment was so small the tuition hardly covered my salary, so the class was not offered again.

To make quality programs work, someone has to require students to take them, so the numbers add up to those trying to make the budget balance.

That reminds me that in my youth I even taught some honors courses for free, i.e. on top of my usual load, just so we could say we offered them, but one cannot forever carry the program on one's back.

Come to think of it, I have been teaching free classes for years now, i.e. when I have free time I have often taught a course that was under-enrolled even though I did not get credit for teaching it. This is still going on by other faculty as well. But it costs a heavy price in time lost from research for the faculty member.

For the faculty member, doing and discussing our research is part of our own "continuing education".

[buffordboy23]

We want students who will try to get to the bottom of things, not merely ones who can compute accurately the area between two curves.

I agree with your view. I think the large problem lies in the fact that many students can do mathematics, but they can't think about mathematics.

Let me try to make this point clearer. From what I have seen with my own educational experience and in working with peers as a returning student, I see that students have greater success with their math/physics courses when it comes to computational problems. Using a previous example from an earlier post, there are only so many ways to compute a line integral, so for a typical problem a student simply needs to recognize the correct form of the given function (i.e. is the function given in cartesian coordinates? parameterized? etc.) and apply the appropriate method to compute the integral. Once one is familiar with the algorithm it is a trivial process to compute line integrals.

In contrast, how would these same students fair in proving the statement, "Show that the zero vector in a vector space is unique." While it may be obvious to the student from experience that the zero vector is unique, how does one show this in a logical proof? This requires the student to think about what the definition of a vector space is and apply it in a novel way. Now, there are common threads when it comes to constructing proofs, but I am not convinced that students are fully aware of such systematic methods.

Looking back on my experience in high school, I was only introduced to solving proofs by direct methods. That is, given the general statement, "If A, then B.," my proof would consist of showing that A implies A1, A1 implies A2,..., An implies B to complete the proof. Most of the proofs I ever did in high school occurred in geometry class using the rigid lock-step method. Looking at my college experience, I am now required to do proofs that required uniqueness, quantifiers, etc; ideas in which I never had formal instruction nor much experience.

While a student may be able to follow the condensed proof given in a textbook by filling in the missing details, many students probably do not think about what caused the author to think in this manner to construct this proof in this specific way. For example, in trying to prove the statement, "The square of an odd integer is an odd integer.", the student should first think, "What exactly is an odd integer?" Answer: 2k+1, where k is an integer. This is what I mean about thinking about mathematics. I own the books "How to Read and Do Proofs" by Solow and "How to Solve it" by Polya, and I must say that these books are indispensible to my current abilities to construct proofs and think mathematically. In general, I think there needs to be a greater emphasis in courses and lectures about how to think about the mathematics rather than doing the mathematics.

Since you have had experience working with public school teachers, have you ever had the opportunity to analyze their abilities to construct proofs? I would guess that many teachers have skills that are inadequate, so our students should have no greater expectations for their abilities. Perhaps, this is something to focus on in future workshop seminars for math teachers.

[harvellt]

Since you have had experience working with public school teachers, have you ever had the opportunity to analyze their abilities to construct proofs? I would guess that many teachers have skills that are inadequate, so our students should have no greater expectations for their abilities. Perhaps, this is something to focus on in future workshop seminars for math teachers.

I think the problem starts well before students get to high school. I read a short article in the New York times a few weeks ago about a state where elementary school teachers were not required to show they had even high school algebra skills. The articles main focus was on a school that decided to higher some math tutors for 5th and 6th graders, not to help struggling kids but to teach the entire math curriculum. The students started to do much better.

[mathwonk]

i try to give my students practice in simple reasoning grounded in their everyday experience; for example:

1. dr. smith's students have placed their hopes in his teaching skills.
2. those who place their hope in weakness are in deep trouble.
3. dr. smith has some of the weakest teaching skills in existence.
conclusion?

[symbolipoint]

i try to give my students practice in simple reasoning grounded in their everyday experience; for example:

1. dr. smith's students have placed their hopes in his teaching skills.
2. those who place their hope in weakness are in deep trouble.
3. dr. smith has some of the weakest teaching skills in existence.
conclusion?

Mathwonk,
You need to modify that presentation or many weaker or less motivated students will simply enroll in courses taught by other Mathematics professors and intentionally avoid taking further courses from you. You run the risk of students making informal counter-recommendations about you as teacher. This could result in less newer students enrolling in your classes. Students who are inadequately conditioned academically will not be able to properly appreciate the meaning of a professor expressing having weaknesses in teaching-skills.

[buffordboy23]

i try to give my students practice in simple reasoning grounded in their everyday experience; for example:

1. dr. smith's students have placed their hopes in his teaching skills.
2. those who place their hope in weakness are in deep trouble.
3. dr. smith has some of the weakest teaching skills in existence.
conclusion?

I understand your approach in trying to teach reasoning skills to students. You should be commended for your efforts, because I see that much of the focus is on teaching students to obtain content knowledge rather than skills to think about the content.

I think students need to learn such thinking skills. Moreover, I believe that there are common themes in thinking about mathematics, as shown through the texts written by Polya and Solow I mentioned in a previous post. These themes seem to be absent from a typical students' education experience, at least from what I have seen. Therefore, students do not have the skills to solve problems which rely upon these themes. Please correct me if I am wrong with this assumption.

[mathwonk]

hows this?

1. the best examples of teachers are those who are held up high by their own students.

2. the best examples of teachers are indeed paragons of greatness.

2. dr smith was carried out of town on a rail held high by his class members.

conclusion?

[mathwonk]

well indeed thinking skills, proof skills, and general argumentatiion are missing from most classes and hence from most students.

we have for some years now tried to remedy this by offering courses in proofs and logic in colleges at the junior/senior level, such as my current course.

this however is apparently too little too late. this need used to be approached in sophomore year of high school by teaching euclidean proof based geometry, but that has all but disappeared from many schools.

so today things are completely upside down. instead of geometry and algebra to 9th and 10th graders with real content, we teach watered down calculus to these same people who now do not even know what the letters QED stand for.

then after they do not learn that, we teach them calculus again in college which again fails because of a lack of algebra skills, and then we try to teach proofs and logic to juniors, and then finally euclidean geometry to seniors and graduate students.

'this hodge podge of remedial teaching is quite a failure all round. for the few lucky ones, we teach them calculus from a good book like spivak in which they are pre -taught algebra and logic and number theory, and calculus with proofs.

this is a brutally intense way to make up in one course for all that is missing beforehand from high school, but is much better than the non honors program. so the best students are very well taught today in some colleges, but the others are much less well taught.

Oh and spivak courses are no,longer offered at such "top" places as Harvard, since their students decline to take it, having had calculus in high school. their beginning course, and that at stanford, is a course from loomis and sternberg or apostol vol 2. this pace quite destroys even some of the most capable but presumably works again for a privileged few.

[mathwonk]

how does a average student cope with a situation like that at harvard today, where the description on their website of math 55 reveals that harvard itself does not offer a course that can prepare its own students to take this course?

i.e. this course can only be attempted by someone who has at the minimum mastered style course. it really is aimed at students who in high school have already taken numerous college courses before heading away to college. so harvard sees no need to try to help prepare the merely talented youngster for its own best courses

this is also true elsewhere. these are really elite schools, since without having prepared in very rarefied circumstances, they do not offer you access to their best courses.

[Count Iblis]

What are some ideas on how to improve this?

Just take a look at how we teach physics at university. I actually learned calculus myself at high school from engineering books. I mastered topics ranging from ordinary differential equations, partial differential equations, Fourier series, Laplace transforms, complex function theory (Cauchy's theorem, residue theorem). All that at age 16, all by myself and I enjoyed learning it.

Thing is that there is no way that I could have mastered these topics from pure math lecture notes that explain everything rigorously. Then the proof that any integral over a closed contour of an analytical function is zero would take ages.

This does not mean that you can leave out all proofs, but I think that by presenting things in a way that we in physics do, you can teach far more math in high school.

[mathwonk]

it really doesn't take long to understand the closed contour fact mathematically.

by the fundamental theorem of calculus, if you are integrating dF along a curve,

the answer is F(b) - F(a) where a,b are the endpoints of the curve.

All analytic differentials f(z)dz have the form dF. QED.

The physical interpretation is that of a force filed that has a potential, i.e. that is conservative.

So in mathematics we have a hypothesis that implies conservative. how does one in physics explain why some fields are conservative and others are not?

I.e. how does one use physics to explain why the integral of f(z)dz around a closed contour not containing any singularities of f, is zero, just because f is analytic?

by the way i agree with you that physics is a great vehicle for learning calculus. it so happens physics is not exactly the most popular subject with many students either. Indeed trying to use physics to illuminate calculus is usually quite challenging since most calculus students in college over the years have known no physics at all.

I used to read to them from galileo, show them the laborious geometric proofs he gave of his results on falling bodies, and try to make the point that calculus renders these easy enough for anyone to derive. sometimes the only thing that got their attention was me jumping up on the table and dropping something, or throwing the chalk in a parabolic arc neatly into the trash can. but we keep trying.

[EC21]

The biggest task I have seems to be helping students learn how to learn. Some fail to come to class, others never look at the notes they take, and many seem not to even open the book.
What are some ideas on how to improve this?



The problem of competitive (un)education

You have competition. It was there before your students entered your classroom and will probably be there after they’ve finished your class. My guess is that you are trying to teach your students good reasoning skills. For every hour of education in your classroom, however, they are likely receiving countless hours being uneducated.

For example: a student turns on the TV and sees Kobe Bryant drinking a bottle of Sprite. Perhaps he identifies the claim as Sprite being desirable. The proof? It’s because someone respectable vouched for it. Following this line of reasoning, a student might think a math proof is whatever you, an expert, said so.

Another example: a government official on TV is claiming that it’s time to go to war. The proof? It’s because now is the time for war. Following this line of reasoning, one might think it justified to prove 1+1=2 because 2=1+1.

Logical fallacies in popular culture or media may seem unrelated to math, but consider the effect that consistently reinforced poor reasoning has on the development of math skills.

Here are some ideas of approach:
1) Offer a token amount of extra credit for each student that prepares an abstract of their goals/interests/hobbies along with their picture and name. This survey lets you know what you’re up against.
2) Perhaps humorously, analyze some logical fallacies in popular media. The goal is to show the power that good reasoning skills have in discerning fallacies (wherever they appear). I think personal discovery is important. If a student has poor reasoning skills, a logical argument may be both ineffective and unpersuasive.


The problem of naïve student view on education

Learning begins with the acceptance of dogma. There is no inherent reason why counting should be 1-2-3 as opposed to 3-1-2. The role of an elementary school teacher is comparatively (and perhaps necessarily) authoritarian. Somewhere around high school and college, the roles change.

I would argue that in college, teachers and students are now engaged in partnerships; each party has their obligations. The problem is that there is no class in school or social cue that appears to facilitate this change.

A proposed solution is to explicitly declare the partnership relationship (which eventually will give rise to mentorship). The goal is to change the view of homework from “punishment” to “duty” and to make apparent the student’s own responsibility in their education. Other cues may be helpful, such as: “You decide your grade, not me.”


The problem of a lack of motivation

Here’s where the class survey would be helpful. Maybe your students are wondering such questions as: How will math make me a better scientist? What does math have to do with my career goals? If a computer can do all these computations, why do I even need to take this class?

Here are some ideas:
1) You could give a periodic digression on the uses of specific topics in industry, sciences, or their relation to higher math.

2) To demonstrate the point that math is useful in daily life, you could ask your students to describe their age without using any numbers. Their answer will probably be in the format a<x<b, where “a” and “b” are shorthand for landmark temporal boundaries. Derivatives are important in driving, knowing when to brake and how much gas to apply. By thinking about the math they already use in their lives, they will discover for themselves how important it is.


The problem of students only doing what is required

Besides developing motivation and a sense of obligation, here are some ideas:
1) Make homework collected but not graded for accuracy. Have it also be worth a negligible amount of points (e.g. 3% of total). The idea is for the experience to be a transition from the (possible ingrained) idea of homework as a necessary teacher appeasement to it being an aid to understanding the material.
2) Quizzes can be given which will use a homework problem, with the exception of slightly changed numbers. Calculator usage is permitted although no partial credit will be given for partial reasoning missing. If time/grading is an issue, you could give quizzes like drug tests, a guaranteed X per semester. I think the important point to be stressed is that quizzes are an important source of feedback, and not just another random hoop to jump through for points.

The problem of poor pre-requisite understanding

I don’t have any ideas to rectify this problem, though I’m interested in hearing them. Theoretically, the student should (and could) go back to learn such pre-requisites. Practically speaking, however, formal U.S. education seems to encourage only progression. Independent Study is always an option but the question is whether or not the student currently has the discipline necessary to set and reach their own goals.


By the way, these opinions are from a community college student who has just completed Multivariable Calculus. I’ve also attempted to pay attention to the various teaching techniques used by my past and present college professors.

-Eric

[Count Iblis]

Mathwonk, yes, I agree. What I meant was that complex differentiability implies Cauchy's theorem (and then you can show that the complex diferentiability implies that the fucntion is analytic). That requires more work.

[symbolipoint]

EC21, from your message, post #148:

Most of what you say is very good. You also did not know what to do about the problem of poor pre-requisite understanding. This is what a school or a department can do about that one: Course registration could require proof of pre-requisite credit, as displayed on transcripts, or as demonstrated on some official/institution assessment test. Also, a DEPARTMENT can apply its own assessment for pre-requisite skills & knowledge regardless of any pre-requisite courses, in case students have not kept those skills & concepts and if those skills & concepts are needed in a course. If pre-requisite knowledge is not adequate, the department can require either student must drop if a full course or more is needed; or student enroll in a targeted remediatory course in order to remain enrolled in the current course - otherwise be administratively dropped from current course. Amazing that students who want to "get through" courses believe that pre-requisite course credit by itself qualifies them for another course, but several weeks after a prerequite course was taken and credit earned, the conditioning from that prerequisite can be lost.

[mathwonk]

I also liked the suggestions from EC21. Time to implement them is of course the next challenge. I am grading exams now and have more questions.

My students have learned from frequent repetition that there are at least two simple properties, each of which imply that a function has an integral, namely it is sufficient for the function to be either continuous, or monotone.

The problem is this fact is useless to many of them, because some students do not seem to know how to recognize monotonicity, nor to understand the difference between "and" and "or", nor between "necessary" and "sufficient".

I.e. immediately after stating that each of the properties above imply the integral exists, some students claim that a function which equals 1 for x between 0 and 1, and equals 2 between 1 and 2, is NOT integrable, "because not continuous", or even "because not monotone".


When asked to state a theorem "with hypotheses" about half seem not to grasp that this means to include the "if" part, the part that tells you when you can use the conclusion.

The "solution" adopted by some is to essentially avoid the use of words, statements of theorems, or arguments of justification for claims. just present computations, and even allow calculators for those, so that none of the rules of computation are even internalized, nor any computational power developed.

To me this is adding to the problem, i.e. that is why many high schools have stopped doing the job of teaching these things- because it is hard to accomplish. But if everyone cops out of trying to teach the use of language and reasoning in discussing concepts, it just gets pushed further and further down the line.

As suggested above, this results in inverted teaching in college. I.e. we continue to teach calculous first, as we did when entering students already knew algebra, geometry and reasoning, but now we teach those prerequisites afterwards.

I.e. calculus is now a 2000 level course, but reasoning and proof is a 3000 level course, and algebra a 4000 level course (this is where students now learn about polynomials and rational numbers), and euclidean geometry is a 5000 level course!

To teach calculus this way, one apparently assumes that students will ignore all parts of the book except the (easier) exercises, never read the explanations, nor even the worked examples, much less the theorems and proofs, and one then spends the class time merely working example problems instead of explaining phenomena and concepts.

But if this model is accepted, wouldn't make more sense to teach from a book like "calculus made easy" or schaum's outline series? instead of stewart or thomas or even better books?

Do you think it could work to re order the courses in college to reflect this change, teaching reasoning, geometry, and algebra first, and calculus later? This would perhaps be resisted by the students who want calculus for other majors, but don't even applied students need to understand how to apply the math correctly?

[mathwonk]

here is a simple conundrum. frequently on a test students have correctly stated in some form, the FTC "part one", i.e. that a continuous function always has an antiderivative and that antiderivative is given by the indefinite integral.

then on the next question, most or all of them have stated that some specific continuous function, like the absolute value function,or e^(x^2), does NOT have an antiderivative, even though all apparently knew these to be continuous.

it is hard for me to communicate with student who sees nothing wrong in making contradictory statements. i can only assume such a student does not know what his statements mean.

It seems many students simply do not realize that the indefinite integral i.e. the integral of f from a to x, is a function of x. Without this they cannot understand the FTC, which says this function is an antiderivative of f, when f is continuous.

Without this understanding, of making a function from a definite integral by letting the endpoint move, calculus is just a process of memorizing rules for areas and volumes without knowing why they work or when they work.

Has anyone succeeded in teaching what the FTC says, and why? does it help to give it a name, like the "moving area function"?

well three students got it right yesterday, so i guess that is progress!

[mathwonk]

the prerequisite problem is raising its thorny head again. in integration, there are 2 methods, substitution and integration by parts. substitution was supposed to be covered in the previous course so we only reviewed it, then taught parts in detail and repeatedly.

on the test most people are getting tedious parts questions correct and many are missing easy substitution questions. these are entering students who took the course previously in high school. maybe we could do some kind of intervention for these students, i.e. stronger than a placement test.

maybe we could have a summer session to help prepare incoming students for college level expectations. but how could we get high school students who think they are above average take a summer remedial course before college?

maybe we should just bite the bullet and admit that essentially all entering high school students are remedial in some way, and simply start out all freshman courses in a remedial way. it seems tricky.

but it is almost impossible now to cover traditional syllabi, when nothing can be assumed as understood from before. one big adjustment seems to be from a learning style where all a student has to do is the required work, to one where the student has to take responsibility for learning the material, doing whatever is personally needed by that student.

i like some form of moonbear's ideas on students presenting or at least talking in small groups. we all know we learned our stuff best when we prepared it to present to a class.

the trick is how to allow students to practice presentations without inflicting a lousy student presentation on the whole class.

i have tried letting students practice the presentation on me in advance but many still did poorly, for one reason or another, usually refusal to practice.

maybe talking in small groups avoids this problem. somehow i feel this uses time in a way that college students should be more grownup than to need, but I am also seeing the need. But ideally it seems these groups should be outside class and supplemental to it

The best model I know of is still Uri Treisman's where there were guided outside problem solving sessions, but we have not been able to provide the support and get the participation for these in the past. maybe we could get a grant for them. this is essentially the vigre model, which is working for grad students, but still in formative stage for undergrads. also it only supports research oriented behavior not basic learning.

[Count Iblis]

but how could we get high school students who think they are above average take a summer remedial course before college?


Let students take voluntary test exams after such courses. If they do well, they pay less tuition fees. They can participate in these exams without following the remedial course, if they think if they don't need it.

[mathwonk]

here's another problem that surprised me: several students think the inverse tangent function is 1/tangent, i.e. cotangent.

the book does write it as tan^(-1), but i explicitly pointed out the possible confusion, and always wrote it as arctan for that reason.

these are again students who have had calculus in high school and passed AP tests high enough to exempt first semester college calc, and do not know what an inverse trig function is. we did not treat them from scratch but have repeatedly used and calculated with them, using the defining property that tan(arctan(x)) = x.

maybe these students are among those who confuse function composition with multiplication. it is hard for me to mentally orient my expectations for students who are supposedly strong and advanced calculus students, but whose gaps in knowledge are those of very weak or beginning precalculus students. i just don't know where to meet them. if a course has certain prerequisites and the students have been placed there it seems natural to assume some of those things.

again i think outside discussion sessions might give more opportunity for random ignorance to surface and be corrected. indeed when i was in college we had problem sections with TA's leading them, and i felt more at ease there, but there is no money now for those. maybe we should go to larger lectures, but with problem sections as well. maybe a lecture will not be much worse if larger. Again, qualified TA's and funds must be found for them.

[mathwonk]

when i look back on my own classroom career as undergrad student, i did not master anything there, but i was fired with enthusiasm and excitement for some of the topics, which i then mastered later.

in this vein i try to show my students some of the connections that i have noticed while teaching the material in their course, such as the link between work and volume.

I.e. if you look at a uniformly massive plane region below the x axis and compute the work done to raise it to the level of the x axis, assuming that is ground level, you get an integrand like yL, where L is the length of the horizontal slice of your region at depth y.

If you think about it, this is the same as the integrand used to compute the volume generated by revolving this region around the x axis, by cylindrical shells, except for a factor of 2pi.

thus if we are doing a work problem with a solid region, such as pumping the water from a swimming pool up to the surface, it follows that this is the same except for a factor of 2pi, as the 4 dimensional volume generated by revolving this solid around the x,y plane, in 4 space!

consequently, if the swimming pool is a hemisphere, and one computes this work, one can obtain the volume of the 4 sphere easily by multiplying by 2pi, which turns out to be pi^2/2 R^4, where R is the radius.

I.e. just as a 3 dimensional ball is generated by revolving a half disc around a line, so a 4 ball is generated by revolving half a 3 diml ball around a plane.

I hope this sort of thing will magnetize some of them to want to understand more mathematics, more effectively than learning to integrate tan^3.

it turns out that the volume of an n diml ball of radius R, equals (2/n)pi R^2, times the volume of an n-2 diml ball of radius R. Maybe some will have fun trying to puzzle out why?

this is essentially archimedes' discovery for the three ball and the one - ball (line segment), since he knew the ball occupied 2/3 the volume of the smallest cylinder containing it. i.e. (2/3)(piR^2)(2R) = (2pi/3)R^2 times (2R) =(4/3)pi R^3.

I guess this sort of thing is more my forte than mindless drill.

[Tac-Tics]

maybe these students are among those who confuse function composition with multiplication.

Function composition is hard because it doesn't come up very much explicitly. The notation for functions also causes problems. I don't think any freshman in college (nor many that graduate) are able to distinguish between the function "f" and the evaluation of f at a point x, "f(x)". In algebra, students learn "y = f(x)", but the it leaves it unclear notationally whether y is a real number or a function. Most students can get the correct answer without being able to make this distinction, but it doesn't lead to a solid understanding.

here's another problem that surprised me: several students think the inverse tangent function is 1/tangent, i.e. cotangent.

Why do they even teach those weird trig functions? I don't even know a geometric situation in where any of the reciprocal trig functions come up. I always found them to be particularly useless, and in high school, I refused to memorize them. I'd rather work out the derivatives with the chain rule at the start of each test.

I do agree that the notation is bad. Personally, I always write out arcblah.


It seems like you're really frustrated with some of your students and their level of preparedness (or competence in some cases). Does what you've written here reflect the majority of students, though? What are the best students' abilities like? What kinds of misunderstandings or gaps in knowledge come up with them?

[buffordboy23]

Mathwonk,

Have you ever consulted the educational literature on teaching mathematics and improving student achievement?

As a former teacher, I had the opportunity to read some articles. Some were beneficial to my effectiveness as a teacher.

[mathwonk]

I have extensively read articles on teaching methods for decades, and tried many different tactics. the definition of "improving achievement" however is not even clear agreed upon. E.g. raising scores on standardized tests is a common goal. the booiks by john saxon were written with this in mind,a nd for a while studies showed they succeeded.

they succeeded however at the price of de emphasizing thinking skills and totally focusing on rote drill. thus student achievement was raised in a narrow even harmful sense, while understanding was completely undermined and sacrificed.

these books were used for years at my son's school until at last they concluded "after using saxon the students didn't understand anything".

my situation is less of someone who is not familiar with the suggestions out there, but of one who has spent some 50 years trying them. some one has said lack of success is about the teacher, but this is not the experience i find. indeed it is almost all about the students in many settings. if we have a teacher we want to promote, it is easy, we just assign that teacher to honors classes with the best students.

these students actually show up, do the work, and appreciate the teachers efforts, and they say so on evaluations. bingo the teacher looks good.

[mathwonk]

here is a specific flaw in algebra the "saxon way", or the mechanical way.

letters in algebra are used properly to represent a range of values. they must be manipulated in a way that would be valid for any of the range of values they may assume.'

hence they are just placeholders for any of a set of values. this is amde clear in jacobs's books, where blank boxes are sued in place of letters at first.


a student who assigns meaning to the letter it does not deserve thinks the same letter must always mean the same thing. e.g. such a student can solve a separable ode

of form dy/dx = 1 + y^2, by rearranging it as dy/(1+y^2) = dx, and integrating both sides.

but if you ask that same student to solve the ode f' = 1 + f^2, he cannot do it. to him the letter f cannot be treated like the letter y.

there is no understanding that a variable is a variable is a variable, no matter what it i called. this loss of the grasp of how to properly use variables creates another gap in understanding for many college students today, who may even have used calculators to "solve algebraic equations" the meaning of the statement represented by the equations is totally lost upon them.

[buffordboy23]

I see. The cloudiness that exists for the definition of what constitutes "improvement" is definitely a problem.

How does your definition of improvement and teacher expectations compare with other colleagues in the university setting?

How about compared to those of organizations like the National Council of Teachers of Mathematics, who define standards for K-12 mathematics education? What about the state of Georgia's standards for math education? Do their standards seem appropriate for college expectations?

[Tac-Tics]

some one has said lack of success is about the teacher, but this is not the experience i find. indeed it is almost all about the students in many settings.

It's a societal thing. Unless they have a personal interest, there is no motivation at all to do it. American society teaches us that math is nerdy, and if you like math, you're socially inept. You can't blame the students. You can't blame your teaching methods either. It doesn't matter if your class is the sturdiest rung in the ladder if the rest of the rungs are damaged.


e.g. such a student can solve a separable ode

of form dy/dx = 1 + y^2, by rearranging it as dy/(1+y^2) = dx, and integrating both sides.

but if you ask that same student to solve the ode f' = 1 + f^2, he cannot do it. to him the letter f cannot be treated like the letter y.

This seems like an excellent technique for testing understanding as opposed to memorization. If the student isn't able to recognize the underlying concept, he or she is in trouble, because you can't apply any technique to solve the problem until you know what kind of problem you're dealing with.


there is no understanding that a variable is a variable is a variable, no matter what it i called.

The idea of a variable is not a simple one. Learning about functional programming languages and formal logic helped out tremendously.

[mathwonk]

I am a bit out of touch with the official definitions of performance and expectations. As I posted earlier, here at Georgia, we have one of the best math and science education departments in the US, indeed it was the ONLY department considered exemplary in a survey of 77 departments nationwide.

Nonetheless the performance of Georgia students on nationwide standardized tests is almost at the bottom of the nation, e.g. in SAT scores. What to make of this?

A long time ago, a friend who was involved in developing materials, asked me to evaluate some of the material then in use for measuring high school teacher qualifications in the state. There was an official list of topics to be covered in schools at various levels, and there was a test to gauge teachers mastery of these.

My role was to evaluate the practice and review materials offered to the teachers who were preparing to take the test. It was simply abysmal.

The syllabus was far too optimistic for one thing. There was every topic in the world on there, and it is completely hopeless to expect any high school teacher to know all that. i still don't know all those topics after 35 years in the field as a professional.

When you see something like that you know it was made up by someone who knows even less than you do about the topics, someone who just took a list and decided that our teachers should know everything anyone might ever want them to know, or we should pretend they do.

then the review materials of course covered only a tiny fraction of those topics since someone had to actually know something abut the topic they were pretending to offer a review of. so the syllabus was a fake.

But the review questions were also sadly inadequate in most cases since they were clearly prepared by people who did not understand in the least the topics they were trying to test. Most of the questions either were unrelated to what they were supposed to test, or were trivial, or were actually wrong. Many of the multiple choice questions contained no correct answers at all, even though the review book said one of them was indeed correct. The reasons offered for the correctness or incorrectness were also wrong.

in reviewing the calculus materials i actually solidified greatly my own understanding of the process of finding volumes, since i needed to really understand it well to recognize wrong questions quickly and to be confident i was in fact right, when the answer book said otherwise.

When you see something like that, you immediately hypothesize that it was prepared by someone who had some kind of tenure in lieu of qualifications. My friend confirmed that there was indeed a grandfathering mechanism in place whereby the teachers already in place were deemed qualified by that fact, and were then asked to test the candidates.


Things have presumably changed enormously since those bad old days.

Nonetheless, there is a lot of politics involved in education, and no matter what we teach our teacher candidates, they still have to go out and please some local school board that likely as not is focused on standardized test scores.

After all thats all we have to go by in many cases. thats all that is telling us our students here perform badly. I had a blessed opportunity once to teach in a local private school to a highly selected class of their best high school students and teach them whatever I wanted out of a book i chose. i taught linear algebra and vector calculus out of marsden and tromba, a book once used at berkeley. and i did it for free.

still the results were mixed. several of those students who took my class because they wanted to, did excellently, went to ivy league schools and obtained phd's in math and physics. others who were there to please their parents hated the class and felt justified afterwards when their college courses at state schools were in fact easier than mine. they thought that proved i was teaching a bad class, because they didn't need such a hard class to prepare them to pass in a mediocre school.

If I hadn't been required to give grades it would have been fine, since the misfits would not have been threatened and the best of the good students would still have worked hard.

l believe that in the introduction to one of the most famous physics texts in existence, by one of the most famous and celebrated professors in history, the feynman lectures, he admits that the actual course he taught was only successful for a few happy natures among his students. teaching is a collaboration between student and teacher, and there has to be an appropriate match for it to work.

[mathwonk]

One of the reasons the UGA math ed dept is so well regarded is they focus on student understanding,a s opposed to rote learning.

in part, they use materials developed by dr. sybilla beckmann that are really quite well done. It is no trivial matter to successfully teach these materials however, but they are widely praised for their potential value in improving learning.

We are interested in identifying people who are dedicated to helping teach these courses and wish to make a profession of developing outstanding teachers. This of course involves taking students who have been taught up to now in the traditional ways, and trying to produce graduates who value understanding, and are committed to teaching for understanding.

If someone wants a graduate math, or math ed degree, focusing on that aspect of the profession, they are invited to apply. Contacting Dr. Beckmann is a good way to begin.

[mathwonk]

here is another specific challenge to teaching basic calculus today, arclength.

the formula is simple enough: integrate sqrt(1 + (y')^2).

the problem is that it is hard to integrate a square root.

there is a trick that is usually used to make doable examples. namely

the simple fact that (a-b)^2 + 4ab = (a+b)^2, can be used if we rig

our arclength example so that (y')^2 has form (a-b)^2 where also ab = 1/4.

I.e. then we get (1 + (y')^2) is a perfect square. the problem is however that it is

almost impossible to give an example that today's students can simplify correctly.


I.e. after carefully explaining this algebraic trick, most in

my classes can still simply not simplify 1 + (y')^2, when y = say x^2/4 - ln(x)/2.

here we get (y')^2 = (x^4 -2x^2 +1)/4x^2, so that adding 1,

changes it to (x^4 +2x^2 +1)/4x^2, a perfect square. this is just too hard.


sometimes i try an easier version, like y = (2/3)(x^2 + 1)^(3/2),

where we get (y')^2 = 4x^4 + 4x^2, but still less than half of a typical class can

see that then 1 + (y')^2 is a perfect square.


I suspect this problem is part of the reason that in the recent book by Rogawski,

arclength is set apart from volume and work,

in a chapter called "further applications of the integral"

as if it is somehow more advanced, and may be skipped.


When a typical calculus student cannot recognize, even with specific instruction,

that (a-b)^2+ 4ab = (a+b)^2, something seems amiss.

[mathwonk]

the persuasive discussion above almost convinces me at last, that it is better to eschew such artificial arclength examples, and focus instead on natural ones, like finding the arclength of a parabola like y = x^2/2, where the difficuloty instead is to integrate

the integrand sqrt(1+x^2), which yields to a trig substitution,plus some tedious integrations by parts.

maybe todays classes will actually find those difficulties more palatable than the algebra of (a±b)^2.

[maze]

I think the parabola example will be much better received, and here's why:

If you use contrived examples based around a trick like using perfect squares, your students will get the wrong impression that finding arc-length is all about perfect squares, when really the two topics are quite unrelated except for in this relatively unimportant class of examples. The example will also seem terribly unmotivated - there are an infinity of different possible curves, and you are choosing a random one for your own seemingly nefarious purposes. It reinforces the negative view that math is all about applying a sequence of memorized manipulations to transform symbols from one form to another.

Better to provide a difficult but well-motivated example. "what is the arc-length of a parabloa"? That's a simple example that anyone would want to know. Some students in the class probably already asked themselves this question. People will naturally work harder if they genuinely want to know the answer.

[Count Iblis]

When a typical calculus student cannot recognize, even with specific instruction,

that (a-b)^2+ 4ab = (a+b)^2, something seems amiss.

Then your students haven't been exposed to math at school enough. I don't think it matters much exactly what they learn, as long as they are working with math formulas and doing non-trivial manipulations they shouldn't have such problems. But math education in school is limited to learning a few specialized techniques and practicing them over and over again, making math in school very boring.

It's a bit like trying to teach language to children by first letting them learn the alphabet then words and then grammar, sentence construction etc. etc.. Only if they have mastered all that perfectly will you think of letting them read books. I think that by that time they would have dropped out and those who haven't would have great difficulties learning to read and write.


In case of math eduction, we need to start changing things in primary school. We need to spend far less time on teaching children arithmetic. Because today we have calculators and teaching things like long division is a complete waste of time. It is useful later, if you learn algebra and want to divide polynomials. Then you also understand why the long division algorithm works at a deep level. Most children in primary school do not really undertand why the algorithm produces the correct answer.

We have to focus more on teaching things that are relevant. Today we work a lot with computers, so it would be a good thing to get rid of a lot of arithmetic in primary school and instead teach logic, computer programming etc. Children then get used to working with undetermined variables. They get to see what they are doing is working when they compile the code they have written.

Then learning algebra will be much easier for them. One can always return to letting them learn more arithmetic later. Then it won't be a trick they have to use, but they can understand much better why it works. Topics like modular arithmetic, Euclid's algorithm, Chinese Remainder theorem etc. should also be taught.

[Count Iblis]

Example of arithmetic using algebra:

If you understand that (N-X)(N-Y) = N(N-X-Y) + XY

You can simplify multiplications in your head. It does not offer any speedup from an algothmic point of view, but you can transform numbers to simpler numbers that your brain can handle more easy.

So, if we have to multiply X, and Y and one of these numbers is close to some round number N, then putting X' = N - X and Y' = N - Y, we have X*Y = N(X - Y') + X' Y'

Suppose then that you want to multiply 89 with 92. Then you choose N = 100, and we have 89*92 = 100*81 + 8*11 = 8188.

You can also use this iteratively. The great advantage of this is then that you can choose your N so that you get numbers that are easy to work with.

Multiplication in the tradional way is useless in practice if the computations you have to do are not simple. You either have a calculator or you are in a place where you quickly need to compute something in your head. Rarely will you be somewhere where you have a desk with paper and pencil and no calculator or computer.

Another example.

87*73

If we take N = 100, then we have:

87*73 = 100*60 + 13*27

If we take N = 30 to compute 13*27 we get:

13*27 = 30*10 + 17*3 = 351

So, 87*73 = 6351

[EC21]

Without this understanding, of making a function from a definite integral by letting the endpoint move, calculus is just a process of memorizing rules for areas and volumes without knowing why they work or when they work.

Has anyone succeeded in teaching what the FTC says, and why?

I am not a teacher and can’t share anything that worked. I can, however, tell you personal and observed difficulties in trying to understand the FTC. You may already be aware of these roadblocks, but if not then perhaps you can come up with some remedies.

The FTC was taught early into my first calculus course. I perceived an integral to simply be the “area under the curve” and the FTC was a coincidentally cute trick to finding it. Why? One reason is because other students told me so. Another was because the book’s statements used a level of abstraction beyond my current level of comprehension and consequently I did not understand them. Additionally, colloquial usage of the word “theory” implies uncertainty (e.g. I have a theory about wheat bread, theory of evolution, etc.) and so I didn’t take the FTC seriously. Finally, I asked around for the applications of calculus and was generally told that it found the area/volume of a variety of “nice” regions.

I also didn’t know what a function was. Some unlearning was involved as I had incorrectly thought f(x) to be an exclusive synonym for y, the dependent variable. Elementary notation increased my confusion. If a function is to be thought of as a sort of taxi between sets, (f o g)(x) seems much less intuitive than ((x)g)f.

[mathwonk]

with regards to jeff foxworthy, you may be an algebraist if ((x)g)f seems natural to you.

this is indeed the algebraists' rule. but they are the apparently only people who refuse to bow to tradition in this matter.

[mathwonk]

some people question whether any theory should be included in a practical calculus course. I still feel some grasp of theoretical aspects is needed to use the material correctly.

here is an example. the integral test for convergence of a series does not work unless the terms a(n) of the series can be extended to functional values a(x), such that not only is the improper integral of a(x) finite (from x=1 to x=infinity), but also the function a(x) has to be decreasing.

yet none of my students bothered to verify the decreasing part when using it, although we gave examples in class of series which diverge although the function has finite integral, when this property does not hold.

one didactic technique for this type of thing was introduced in the book in another test, the alternating test, which the book called the "3 condition test", to remind the student how many hypotheses there are.

this trick has limited use however, since it is hard to call every theorem the "n hypotheses theorem".

[mathwonk]

if you have a sudden loss of hearing or ability to concentrate, or your wife tells you you are becoming really boring, stop taking calculus immediately (due credit to cialis ads).

[mathwonk]

after all is said and done this semester, one overall problem I still see is the tension between a professor as teacher (or one who gives insight), and as "personal trainer".

For the student, it is crucial to attend class and do the recommended work regularly, to read independently, formulate and ask questions, and visit office hours for help with difficulties. It is a truism in the profession that the most successful student is the one who displays these behaviors, not the one with the quickest mind.

But what to do when faced with the realization that many students do not do all or sometimes any of these things? Then the professor feels pressure to "force" compliance with these good habits, i.e. to become a personal trainer rather than a teacher. This occurs especially in high school when teaching is measured by student performance on standardized tests, and it is hard for many students to begin to behave differently when entering college.

In the short run, indeed the techniques of personal trainers get better results on tests.

In the long run, however, do these techniques allow students to delay taking responsibility for their own learning? How can one enable sincere but naive students to acquire basic information, in spite of their poor study habits, and yet also encourage students to begin to assume responsibility for learning?

The professor is essential for sharing insight acquired over decades into a difficult topic, but for acquiring skill at a technique that has already been explained, only self discipline is needed. How much practice is it essential for the professor to enforce, and how much should be expected from the learner?

In a nutshell, if it takes three repetitions for a skill to become habit, some students will expect it to be repeated three times in class. But in college it is more usual to present it once, and then expect the student to do the other two reps at home. How to get this lesson across, so the professor's time can be spent more profitably explaining new topics, and deeper aspects of the subject?

One approach is to do the repetitions in class, making time for them by omitting the deeper aspects of the subject. This is called dumbing down the course. Creative ways to avoid this approach are needed.

Perhaps best is the time honored one of making advance assignments, then sending random students to the board to display what has been done. This is very time consuming, but experience shows it is helpful for many students at all levels, even into graduate school. Indeed since even professors use this method in learning seminars, perhaps the realistic thing is to reduce the syllabus of undergrad courses to allow more time for this activity.

[Troponin]

To put my very, very late for the seminar point of view here...

I am returning to school at the age of 29, 7 years after completing a degree in kinesiology, to study math and physics.

I spent my entire life believing I hated math and was horrible with numbers. I was told I wouldn't be good at math because I don't have the focus to follow the structure and order that maths require.

I was a good student throughout my life, and even did well in math up through geometry. The following school year in pre-algebra (this was early high school), I grew to dislike mathematics more than any other subject.

It seemed that every single thing we did was an arbitrary rule to follow for no other reason than.....that is what you did.
Any question of "why" a rule was as it is was quickly hushed to make more time for practicing arbitrary rules.

Any question that didn't follow the order and rule set for that type of problem was marked wrong....regardless of the correctness of the answer.

I could continue with examples and frustrations, but I'm sure it will be nothing that hasn't already been covered here.

As a child, I loved science and drawing. I loved reading about the stars, looking through a microscope....even loved doing those math books you get at the drug store. I had always wanted to be an engineer....for no other reason than I was once told they were scientists that "drew things." lol

When ever I would mention that career goal to an adult, they would undoubtably give me the same response, "you know..that's a LOT of math...."

After pre-algebra, I avoided math classes for the rest of my studies. I took no math electives that weren't absolutely required in college and accepted that I hated math.


Years later, I got the urge to read up on cosmology and astronomy again...like I used to love when I was a kid. After about a dozen "made for the public" physics books, I decided I should learn some math so I could read something with a bit more detail.

I purchased a "teach yourself calculus" book with the belief that it would be a torturous process I had to slave through for the better good of my reading.

Turns out I LOVED it. When left to my own, I could spend the time learning the theory of each concept and seeing what it could do from top to bottom. I poured through the differential portion of that book in about a week.

I'm now back in school and should have a second degree with Applied Mathematics major and Physics minor completed next winter. I hope to move on to a PhD program in Physics (hopefully theory...) following that.

I don't have any real advice to offer the situation, but it frustrates me deeply to look back at my grammar school years and remember how math was presented to me.

Math was for the socially inept nerds....organized accountants that love nothing more than spending their day doing long division. I'm happy to know that I'll be able to present mathematics to my children in a manner that allows them to see what it really is (or what I think it is). But, it frustrates me to know there are probably many more science nerds like me that were taught to hate mathematics.

[Tobias Funke]

Troponin, it's nice to read stories like yours and I only wish they were more common.

I don't have any real advice to offer the situation, but it frustrates me deeply to look back at my grammar school years and remember how math was presented to me.

Yeah, it's sad when my 9th graders tell me that their teachers marked them wrong if they left their answer as 3/2 instead of 1 1/2, or how they were forced to write 2+(-3) instead of 2-3, for example. Stuff like this is very common in elementary and middle schools and I'm glad you recovered from it. Good luck!

[Troponin]

Troponin, it's nice to read stories like yours and I only wish they were more common.



Yeah, it's sad when my 9th graders tell me that their teachers marked them wrong if they left their answer as 3/2 instead of 1 1/2, or how they were forced to write 2+(-3) instead of 2-3, for example. Stuff like this is very common in elementary and middle schools and I'm glad you recovered from it. Good luck!

Oh yes.....my first semester back in school involves a story about that. I went up to my professor and asked a situation similar to that....he looked at me like I was insane. To my relief, he said "it doesn't matter...it's the same number...I don't know what you're asking?!?"

I'm sure he was even more confused when I seemed extremely happy with his non-answer...lol

[mbisCool]

When i was younger my teacher marked all of my answers wrong because instead of putting a semi colon between my answer and restriction for what x couldn't equal, i simply put a space...

She also marked me wrong when the end of my square root symbol didnt fully extend over the last number. :/

[Troponin]

When i was younger my teacher marked all of my answers wrong because instead of putting a semi colon between my answer and restriction for what x couldn't equal, i simply put a space...

She also marked me wrong when the end of my square root symbol didnt fully extend over the last number. :/


I have a differential equations professor that is FANTASTIC. He doesn't care about ANY of that. He wants you to "understand" what is going on....and almost prefers you to not follow the "plug and chug" list of operations for each type of solution.
He's a very difficult professor so a lot of students seem to really dislike him, but I absolutely love him.

All the arbitrary accounting style order and rules that I've always felt had more to do with memorization than understanding the materials are considered just that.....arbitrary memorization that doesn't prove if you understand the material.

[descendency]

What are some ideas on how to improve this?
More abstinence from people who shouldn't be parents!

By the time you get these students in Calculus I, you have gotten a student who has developed years of **** poor mathematics. I can't say I was a great mathematician taking calculus I, but I can say I knew what the basic terminology was.

Knowing what a plane, a denominator, and other very simple concepts are is fundamental to have any hope of passing calculus.

Unfortunately, teaching fundamentals and refusing to work with students who won't meet you on a fundamentals level is the only hope you have.

If they don't understand what a number line means how will they understand a delta-epsilon proof? or a limit? or handedness? or derivatives? or integrals? or techniques of integration? or optimization? or applications of integration? or anything else?

edit: As for marking off points for sloppy work, you deserve them. When an instructor sets out a standard, you deserve points off for not being able to follow a standard. Half-***ing it is just not acceptable nor should it be. Not drawing a division bar long enough shows how little you care for the work you are doing. You feel your time is being wasted. If that's the case, don't waste your instructors time with your test/homework.

If the instructor themself can not follow a standard though, then you have a gripe. However, drawing arrowheads on your graph to indicate direction of increasing is not too much to ask. Or labeling the axes.

[uman]

I would consider it unreasonable that a teacher would mark off for using a semicolon instead of a space.

[Tobias Funke]

Teachers should only set reasonable standards. What if a teacher wanted you to write a...@b instead of a+b for the semester? Sure it's an extreme example, but it's no more silly than requiring 1.5 instead of 3/2 (unless the point of the exercise is to get practice with decimals). Descendency, it seems like you're thinking of something like the difference between \sqrt{a}b and \sqrt{ab}, which definitely deserves some correcting. I'm talking about much more trivial things.

Also note that these experiences all seem to stop in college. Professors aren't going to care if you write A-B or A\setminus B, but too many 1-12 teachers would and it's just plain stupid sometimes.

Actually I guess the equivalent in college would be those annoying computer hw sites that only accept the answer in a certain form that doesn't seem to be any better or neater than your own. Luckily I've never had to use them, but I tried helping a friend of mine and was very frustrated.

I feel like ending this on a positive note though. Today one of my students got a 100 on a math test for the first time in a year, and another student said "this is actually kinda fun once you get it" about solving systems of equations! That just makes the day so much better

[qwerty2x]

We had our year 12 end of year exam for specialist math(for the whole state) and we were allowed to bring in a bound reference. Everyone came in holding a large book with LOTS of papers glued into it that had problems and questions with the working out, while i came in with a lousy one-paged A4 sheet of paper with almost nothing on it as i didnt know what to write on it(its a math exam what am i suppose to put on it??).
Anyway the exam started and im doing the problems, i look up and i see people flicking through their papers "trying to find a similar question" to what was being asked.

I ended up getting the highest in the class and in the top 7% of the state, i guess the way the other students learn the material is really the problem these days(mind you their parents are payed **** loads for private tutiton).

would you say that the rate of drop-outs for engineering students in higher than that of most other disiplines?

[symbolipoint]

qwerty2x, you made a nice observation about test-taking and possibly study methods, and then after that, you made a statement which seems unrelated. Readers may have clearer responses or comments if you would show a transition to that last statement.

One reason ignoring quantity for why any university or college student would drop from their formal education is to work, do an employed job, or start a business. You are probably trying to address something opposite to that. Engineering may be difficult to study well. Students might drop because they need to work to earn money/more money, or change major field due to difficulty of coursework, or drop intending to return but never do (but possibly work for many years after dropping from Engineering study).

YOU might be either more talented than the other test takers, or you have better study methods than they, or you may have studied longer and harder than they. Good Work!

EDIT: How much did this test involve Calculus? Could you tell us which topics from Calculus were in the test's content? Did the test rely on multiple choice answers, were showing written steps a main feature of your responses to the questions? Were proofs involved? Were essays involved? These are useful questions in this thread-topic since this could relate to the title, "teaching calculus today in college".

[qwerty2x]

the exam was made up of 22 multiple choice questions and 5 extended answer question.
Here is a link to part 2 of the exam(cant find part 1)(not uploaded by me).

http://trinon.info/exams/VCAA_2008_Specialist_Mathematics_Exam_2.pdf

EDIT:yeh sorry that statement does sound unrelated, just wanted to get a feel of the difficulty of engineering at uni.

[buffordboy23]

the exam was made up of 22 multiple choice questions and 5 extended answer question.


There are many questions on that test (particularly complex variables and differential equations) for which my high school education would not have prepared me. And, I was an accelerated mathematics student in my school, which was a very small school in a somewhat rural setting.

Who takes this test? Where is it administered? What is its purpose?

[qwerty2x]

There are many questions on that test (particularly complex variables and differential equations) for which my high school education would not have prepared me. And, I was an accelerated mathematics student in my school, which was a very small school in a somewhat rural setting.

Who takes this test? Where is it administered? What is its purpose?

Every year 12 student that takes the subject in victoria(approx 6000 people i think, small compared to other subjects) take this subject, it is like advanced math and i think its the second hardest math that a year 12 can take in victoria. administrators are VCAA.

Part 1 of the exam was much harder even tho it was only 10 questions.

[kbaumen]

I feel like ending this on a positive note though. Today one of my students got a 100 on a math test for the first time in a year, and another student said "this is actually kinda fun once you get it" about solving systems of equations! That just makes the day so much better

I definitely agree to the last sentence. Of course, I'm not a teacher, just a 12th grader but sometimes (quite often actually) I help other students from my class and it really feels nice to hear them say "wow, now that you explain, it seems quite interesting, hey if I substitute this with that, you get this and that's how you prove that, so this is another application of that, wow..." I can imagine, that being a teacher even if 99 of your students think that you suck as much as the subject does, it's still worth sucking it up if later you get a "thanks a lot, this is very interesting" from one student.

[Moonbear]

Teachers should only set reasonable standards. What if a teacher wanted you to write a...@b instead of a+b for the semester? Sure it's an extreme example, but it's no more silly than requiring 1.5 instead of 3/2 (unless the point of the exercise is to get practice with decimals).

In elementary school, when they make you do exercises like that, it is indeed the objective of the exercise to practice manipulating fractions or converting fractions to decimal points. It's not just a teacher being nitpicky, it's a teacher trying to ensure the students have the fundamentals correct. Part of that is making sure a student understands that 3/2 is the same as 1 1/2 and isn't confusing it with 2/3. When learning fractions, confusing the numerator and denominator is commonplace, so forcing students to do added steps that demonstrate they comprehend the distinction is necessary. The point of homework and exams is for the student to demonstrate they have mastered whatever topic they are learning, not that they have provided some reasonable semblance of an effort enough to get away with others guessing what they are thinking. And, when so many mistakes in math happen because of carelessness and sloppy handwriting, it's good for teachers to enforce those rules early so students develop careful habits for when the math gets harder. It's the same reason teachers grade notebooks; it makes students develop good habits.

[maze]

But it also causes kids to feel like math is nothing more than pedantic manipulation of symbols according to arbitrary rules. This couldn't be further from the truth, and it directly contributes to the dislike of math by the general population.

[buffordboy23]

But it also causes kids to feel like math is nothing more than pedantic manipulation of symbols according to arbitrary rules. This couldn't be further from the truth, and it directly contributes to the dislike of math by the general population.

Are you suggesting that making students accountable for the quality of their work and meeting teacher expectations is the reason for the dislike of math?

In adding to what Moonbear and Tobias Funke stated previously, I think the most important question to ask regarding this matter is, "Does such and such requirement align well with the lesson or overall course objectives?" For third or fourth graders, requiring them to carry out each step when it comes to fractions is acceptable in my opinion for the reasons Moonbear mentioned. But for a typical eighth grader, forget about it since this skill should have already been obtained.

I agree with what you said on why math is disliked by the general population: it's nothing more than pedantic manipulation of symbols according to arbitrary rules (and a nice catchy mnemonic tune every now and then). I think the fundamental and underlying reason for this is the teacher and the methods of instruction employed. Usually, these instructional methods are simply the pedantic manipulation of symbols.

[vsheng]

I am a student in university, recently i am facing problem to deal with my calculus lecturer,
well, actually they are certain similiar point with your problem with the problem between me and my lecturer.....

"Many never ask questions, and those who do, often ask things that could be found immediately by looking them up in the index of the book. "

Most of my friend do not ask question because they cant even understand what the lecturer talking about, for those understand who are able to ask question, they are often in a ;hald understand' condition, when they are in that condition, dont expect them to think normally as a normal person.


"Questions more often focus on "what will be tested?" instead of how to understand what has been taught."

Every student are worry and emphasing the result of their academic, if not, they are not actually concern about their future..

"Everyone seems to have taken calculus in high school, but most also seem not to know anything about algebra or geometry or trigonometry. With the advent of calculators some also do not know simple arithmetic, like how to multiply two digit numerals. (I have had students who had to add up a column of thirteen 65's on a test, apparently not knowing how to multiply 13 by 65.)
Many think that having taken a subject "2 years ago" is a valid excuse to have forgotten the material, and to expect the teacher to reteach the prerequisites. Appparently no one ever dreams of reviewing the prerequisites before the course starts. Books like "Calculus for cretins" are apparently more popular than books like "Calculus for science majors"."

Student have to take a lot of subject beside calculus, and they have to join activities too,
further more, as a student, when they are in holiday, many of them will just let their mind relax in a this short period,and they of sourse, they do not study the previous subjest that they have taken...so, sometimes if they forgot some important steps, it does not show that they are not a good student...

[Tobias Funke]

snip

I agree with you since you were specifically talking about elementary school. But what I'm seeing with my high school freshmen isn't comprehension and easy conversion between number representations. They're just confused. They didn't seem to understand that 22/5 is a number and they insisted on changing it to 4 2/5 or even wasting time doing the long division. Their previous teachers have forced them to write fractions in a certain way and it was hard for them to change. Like maze said, I'm telling them one thing in opposition to their middle school teachers, probably leaving them to stop thinking about the subject and just following my orders.

They're improving their abilities to memorize facts which to them are meaningless and regurgitating on their quizzes. That's not the kind of teaching I had in mind and I'm not sure I'm qualified to teach the low level math that they need. I chose high school for a reason, after all. Next year I'm going to ask to separate the students in the first week or so into algebra 1 and pre-algebra. Whether they officially do so or not, if I teach the same class next year it's going to be a reintroduction to fractions, half eaten pies and all, for about 3 months.